<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c4">IT1</font>, <font color="Maroon" title="c5">IT2</font> be    <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>assume </b><b>that </b><br/><a NAME="E1:59_1_2"/><i><font color="Green" title="E50">A28</font></i>: 
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>  st <font color="Olive" title="b1">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> holds <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> holds  <font color="Maroon" title="c4">IT1</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="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> <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>
 <b>and </b><br/><a NAME="E2:59_1_2"/><i><font color="Green" title="E51">A29</font></i>: 
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>  st <font color="Olive" title="b1">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> holds <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> holds  <font color="Maroon" title="c5">IT2</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="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> <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><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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="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="E2:59_1_2_1"/><i><font color="Green" title="E52">A30</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="E3:59_1_2_1"/><i><font color="Green" title="E53">A31</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">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="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F2</font> &amp; <font color="Maroon" title="c6">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>thus </b><font color="Maroon" title="c4">IT1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">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"><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#E1:59_1_2"><i><font color="Green" title="E50">A28</font></i></a>, <a class="txt" href="isocat_2.html#E2:59_1_2_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="isocat_2.html#E3:59_1_2_1"><i><font color="Green" title="E53">A31</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">IT2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E2:59_1_2"><i><font color="Green" title="E51">A29</font></i></a>, <a class="txt" href="isocat_2.html#E2:59_1_2_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="isocat_2.html#E3:59_1_2_1"><i><font color="Green" title="E53">A31</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:59_1_2"/>
<font color="Maroon" title="c4">IT1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">IT2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
