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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span><br/> for <font color="Olive" title="b3">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Olive" title="b1">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Olive" title="b2">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Olive" title="b3">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Olive" title="b1">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Olive" title="b2">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Olive" title="b3">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<div><i><font color="Green" title="E50">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span>;<br/><a NAME="E1:59_1_1_1"/><i><font color="Green" title="E51">A2</font></i>: 
( the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c6">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>, <font color="Maroon" title="c7">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font><b> such that </b><br/><a NAME="E3:59_1_1_1"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F2</font>
 <b>and </b><br/><a NAME="E4:59_1_1_1"/><i><font color="Green" title="E53">A4</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/><a NAME="E5:59_1_1_1"/>
(  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F2</font> &amp;  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F2</font> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="isocat_1.html#T27" title="ISOCAT_1:th.27">ISOCAT_1:27</a></i>;<br/><a NAME="E6:59_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c5">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c7">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c5">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c7">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E1:59_1_1_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="nattra_1.html#T35" title="NATTRA_1:th.35">NATTRA_1:35</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">g</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c5">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c7">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c5">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c7">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span> <b>by </b><i><a class="ref" href="cat_2.html#T36" title="CAT_2:th.36">CAT_2:36</a></i>;<br/><b>take </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g</font>;<br/><b>thus </b><a NAME="E8:59_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c9">g</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">G1</font>, <font color="Maroon" title="c11">G2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>;<br/><b>let </b><font color="Maroon" title="c12">t</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">G1</font>,<font color="Maroon" title="c11">G2</font>;<br/>



<b>assume </b><a NAME="E1:59_1_1_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">G1</font>,<font color="Maroon" title="c11">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>

<a NAME="E2:59_1_1_1_1"/><b>then </b><i><font color="Green" title="E54">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">G1</font>,<font color="Maroon" title="c11">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">s</font> )
 
<b>by </b><i><a class="txt" href="isocat_2.html#E4:59_1_1_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E3:59_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c10">G1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F1</font> &amp; <font color="Maroon" title="c11">G2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E4:59_1_1_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c10">G1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c11">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c12">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c10">G1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c11">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c12">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="isocat_2.html#E2:59_1_1_1_1"><i><font color="Green" title="E54">A5</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">IT</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span><b> such that </b><br/><a NAME="E3:59_1_1"/><i><font color="Green" title="E51">A6</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="isocat_2.html#E1:59_1_1"><i><font color="Green" title="E50">A1</font></i></a>);<br/></i>
<a NAME="E4:59_1_1"/>
<font color="Maroon" title="c4">IT</font> is    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of  <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:59_1_1_2"/><i><font color="Green" title="E52">A7</font></i>: 
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E2:59_1_1_2"/><i><font color="Green" title="E53">A8</font></i>: 
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<b>thus </b><a NAME="E3:59_1_1_2"/>
for <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span>  ex <font color="Olive" title="b2">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span> st <font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b2">d</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D1" title="ISOCAT_1:def.1">ISOCAT_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">c</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span> <b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self" title="ISOCAT_2:th.8">Th8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">d1</font> =  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self" title="ISOCAT_2:th.8">Th8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">d2</font> =  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self" title="ISOCAT_2:th.8">Th8</a></i>;<br/>
<b>take </b>
<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c7">d1</font>,<font color="Maroon" title="c8">d2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
;<br/>

<a NAME="E4:59_1_1_2_1"/>
 <a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="isocat_1.html#T38" title="ISOCAT_1:th.38">ISOCAT_1:38</a></i>;<br/>
<a NAME="E5:59_1_1_2_1"/><b>then </b><i><font color="Green" title="E54">A9</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c7">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E6:59_1_1_2_1"/>
 <a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="isocat_1.html#T38" title="ISOCAT_1:th.38">ISOCAT_1:38</a></i>;<br/>
<a NAME="E7:59_1_1_2_1"/><b>then </b><i><font color="Green" title="E55">A10</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c8">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E8:59_1_1_2_1"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c6">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">c</font></span>)</span> = 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c7">d1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c8">d2</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="isocat_2.html#E5:59_1_1_2_1"><i><font color="Green" title="E54">A9</font></i></a>, <a class="txt" href="isocat_2.html#E7:59_1_1_2_1"><i><font color="Green" title="E55">A10</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c7">d1</font>,<font color="Maroon" title="c8">d2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#T41" title="CAT_2:th.41">CAT_2:41</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:59_1_1_2"/>
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span> holds <br/> ( <font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c6">F1</font>, <font color="Maroon" title="c7">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>, <font color="Maroon" title="c8">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font><b> such that </b><br/><a NAME="E2:59_1_1_2_2"/><i><font color="Green" title="E54">A11</font></i>: 
<font color="Maroon" title="c6">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">F2</font>
 <b>and </b><br/><a NAME="E3:59_1_1_2_2"/><i><font color="Green" title="E55">A12</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/>
<a NAME="E4:59_1_1_2_2"/>
 <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F2</font>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E2:59_1_1_2_2"><i><font color="Green" title="E54">A11</font></i></a>, <a class="ref" href="isocat_1.html#T27" title="ISOCAT_1:th.27">ISOCAT_1:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">f1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c8">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <b>by </b><i><a class="txt" href="isocat_2.html#E1:59_1_1_2"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="nattra_1.html#T35" title="NATTRA_1:th.35">NATTRA_1:35</a></i>;<br/>
<a NAME="E6:59_1_1_2_2"/>
 <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F2</font>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E2:59_1_1_2_2"><i><font color="Green" title="E54">A11</font></i></a>, <a class="ref" href="isocat_1.html#T27" title="ISOCAT_1:th.27">ISOCAT_1:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">f2</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c8">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <b>by </b><i><a class="txt" href="isocat_2.html#E2:59_1_1_2"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="nattra_1.html#T35" title="NATTRA_1:th.35">NATTRA_1:35</a></i>;<br/>
<a NAME="E8:59_1_1_2_2"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F1</font> &amp;  <a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="isocat_1.html#T38" title="ISOCAT_1:th.38">ISOCAT_1:38</a>, <a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>;<br/>
<a NAME="E9:59_1_1_2_2"/><b>then </b><i><font color="Green" title="E56">A13</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c9">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F1</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E10:59_1_1_2_2"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F1</font> &amp;  <a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="isocat_1.html#T38" title="ISOCAT_1:th.38">ISOCAT_1:38</a>, <a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>;<br/>
<a NAME="E11:59_1_1_2_2"/><b>then </b><i><font color="Green" title="E57">A14</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F1</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E12:59_1_1_2_2"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c6">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1_2_2"><i><font color="Green" title="E55">A12</font></i></a>, <a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> = 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c9">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="isocat_2.html#E9:59_1_1_2_2"><i><font color="Green" title="E56">A13</font></i></a>, <a class="txt" href="isocat_2.html#E11:59_1_1_2_2"><i><font color="Green" title="E57">A14</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c9">f1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">f2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#T41" title="CAT_2:th.41">CAT_2:41</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c10">f2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="cat_2.html#T38" title="CAT_2:th.38">CAT_2:38</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="isocat_2.html#E3:59_1_1_2_2"><i><font color="Green" title="E55">A12</font></i></a></i>
;<br/><br/>

<a NAME="E14:59_1_1_2_2"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F2</font> &amp;  <a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="isocat_1.html#T38" title="ISOCAT_1:th.38">ISOCAT_1:38</a>, <a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>;<br/>
<a NAME="E15:59_1_1_2_2"/><b>then </b><i><font color="Green" title="E58">A15</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c9">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F2</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c7">F2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E16:59_1_1_2_2"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F2</font> &amp;  <a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="isocat_1.html#T38" title="ISOCAT_1:th.38">ISOCAT_1:38</a>, <a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>;<br/>
<a NAME="E17:59_1_1_2_2"/><b>then </b><i><font color="Green" title="E59">A16</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F2</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c7">F2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E18:59_1_1_2_2"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F2</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1_2_2"><i><font color="Green" title="E55">A12</font></i></a>, <a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> = 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c9">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="isocat_2.html#E15:59_1_1_2_2"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="isocat_2.html#E17:59_1_1_2_2"><i><font color="Green" title="E59">A16</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c9">f1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">f2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#T41" title="CAT_2:th.41">CAT_2:41</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c10">f2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="cat_2.html#T38" title="CAT_2:th.38">CAT_2:38</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="isocat_2.html#E3:59_1_1_2_2"><i><font color="Green" title="E55">A12</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span>;<br/>

<b>assume </b><a NAME="E5:59_1_1_2"/><i><font color="Green" title="E54">A17</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">F1</font>, <font color="Maroon" title="c8">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>, <font color="Maroon" title="c9">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c8">F2</font><b> such that </b><br/><a NAME="E7:59_1_1_2"/><i><font color="Green" title="E55">A18</font></i>: 
<font color="Maroon" title="c7">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">F2</font>
 <b>and </b><br/><a NAME="E8:59_1_1_2"/><i><font color="Green" title="E56">A19</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F2</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c8">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">G1</font>, <font color="Maroon" title="c11">G2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>, <font color="Maroon" title="c12">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">G1</font>,<font color="Maroon" title="c11">G2</font><b> such that </b><br/><a NAME="E10:59_1_1_2"/><i><font color="Green" title="E57">A20</font></i>: 
<font color="Maroon" title="c10">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">G2</font>
 <b>and </b><br/><a NAME="E11:59_1_1_2"/><i><font color="Green" title="E58">A21</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">G1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G2</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">G1</font>,<font color="Maroon" title="c11">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">t</font> = <font color="Maroon" title="c12">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c11">G2</font> <b>by </b><i><a class="txt" href="isocat_2.html#E5:59_1_1_2"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="isocat_2.html#E8:59_1_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="isocat_2.html#E11:59_1_1_2"><i><font color="Green" title="E58">A21</font></i></a></i>;<br/>
<a NAME="E13:59_1_1_2"/><i><font color="Green" title="E59">A22</font></i>: 
<font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c9">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c9">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="isocat_2.html#E8:59_1_1_2"><i><font color="Green" title="E56">A19</font></i></a></i>;<br/>
<a NAME="E14:59_1_1_2"/>
(  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c8">F2</font> &amp;  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c8">F2</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c11">G2</font> )
 
<b>by </b><i><a class="txt" href="isocat_2.html#E5:59_1_1_2"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="isocat_2.html#E7:59_1_1_2"><i><font color="Green" title="E55">A18</font></i></a>, <a class="txt" href="isocat_2.html#E8:59_1_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="isocat_2.html#E10:59_1_1_2"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="isocat_2.html#E11:59_1_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="isocat_1.html#T27" title="ISOCAT_1:th.27">ISOCAT_1:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">g1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c8">F2</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c11">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c15">f1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c9">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <b>by </b><i><a class="txt" href="isocat_2.html#E1:59_1_1_2"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="nattra_1.html#T35" title="NATTRA_1:th.35">NATTRA_1:35</a></i>;<br/>
<a NAME="E16:59_1_1_2"/>
(  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c8">F2</font> &amp;  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c8">F2</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c11">G2</font> )
 
<b>by </b><i><a class="txt" href="isocat_2.html#E5:59_1_1_2"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="isocat_2.html#E7:59_1_1_2"><i><font color="Green" title="E55">A18</font></i></a>, <a class="txt" href="isocat_2.html#E8:59_1_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="isocat_2.html#E10:59_1_1_2"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="isocat_2.html#E11:59_1_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="isocat_1.html#T27" title="ISOCAT_1:th.27">ISOCAT_1:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">g2</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c8">F2</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c11">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c17">f2</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c9">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <b>by </b><i><a class="txt" href="isocat_2.html#E2:59_1_1_2"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="nattra_1.html#T35" title="NATTRA_1:th.35">NATTRA_1:35</a></i>;<br/>
<i><font color="Green" title="E60">A23</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c14">g1</font> = 
 <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c8">F2</font>
<b>by </b><i><a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c15">f1</font>
<b>by </b><i><a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>
;<br/>
<i><font color="Green" title="E61">A24</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c16">g2</font> = 
 <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c8">F2</font>
<b>by </b><i><a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c17">f2</font>
<b>by </b><i><a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>
;<br/>
<i><font color="Green" title="E62">A25</font></i>: <font color="Maroon" title="c14">g1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c15">f1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c11">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c13">t</font></span>)</span> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <span class="p3">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c9">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="nattra_1.html#T42" title="NATTRA_1:th.42">NATTRA_1:42</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c11">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c9">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E5:59_1_1_2"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="isocat_2.html#E7:59_1_1_2"><i><font color="Green" title="E55">A18</font></i></a>, <a class="txt" href="isocat_2.html#E8:59_1_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="isocat_2.html#E10:59_1_1_2"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="isocat_2.html#E11:59_1_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="isocat_1.html#T32" title="ISOCAT_1:th.32">ISOCAT_1:32</a></i>
;<br/>
<i><font color="Green" title="E63">A26</font></i>: <font color="Maroon" title="c16">g2</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c17">f2</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c11">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c13">t</font></span>)</span> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <span class="p3">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c9">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="nattra_1.html#T42" title="NATTRA_1:th.42">NATTRA_1:42</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c11">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c9">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E5:59_1_1_2"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="isocat_2.html#E7:59_1_1_2"><i><font color="Green" title="E55">A18</font></i></a>, <a class="txt" href="isocat_2.html#E8:59_1_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="isocat_2.html#E10:59_1_1_2"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="isocat_2.html#E11:59_1_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="isocat_1.html#T32" title="ISOCAT_1:th.32">ISOCAT_1:32</a></i>
;<br/>
<a NAME="E22:59_1_1_2"/>
<font color="Maroon" title="c6">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c11">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c9">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E5:59_1_1_2"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="isocat_2.html#E8:59_1_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="isocat_2.html#E11:59_1_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="nattra_1.html#T42" title="NATTRA_1:th.42">NATTRA_1:42</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c5">f</font></span>)</span> = 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">g1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c15">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">g2</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c17">f2</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="isocat_2.html#E20:59_1_1_2"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="isocat_2.html#E21:59_1_1_2"><i><font color="Green" title="E63">A26</font></i></a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c14">g1</font>,<font color="Maroon" title="c16">g2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c15">f1</font>,<font color="Maroon" title="c17">f2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E18:59_1_1_2"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="isocat_2.html#E19:59_1_1_2"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="cat_2.html#T39" title="CAT_2:th.39">CAT_2:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="isocat_2.html#E5:59_1_1_2"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="isocat_2.html#E8:59_1_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="isocat_2.html#E11:59_1_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="isocat_2.html#E13:59_1_1_2"><i><font color="Green" title="E59">A22</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">IT</font> = <font color="Maroon" title="c4">IT</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of  <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span> ;<br/>
<b>take </b>
<font color="Maroon" title="c5">IT</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">F1</font>, <font color="Maroon" title="c7">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>;<br/>

<b>assume </b><a NAME="E6:59_1_1"/><i><font color="Green" title="E52">A27</font></i>: 
<font color="Maroon" title="c6">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">F2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">t</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font>;<br/>

<b>set </b><font color="Maroon" title="c9">f</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E7:59_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E6:59_1_1"><i><font color="Green" title="E52">A27</font></i></a>, <a class="ref" href="nattra_1.html#T35" title="NATTRA_1:th.35">NATTRA_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">f</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span> <b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c5">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font>

<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c8">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c8">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:59_1_1"><i><font color="Green" title="E51">A6</font></i></a></i>
;<br/>


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