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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c5">F2</font>, <font color="Maroon" title="c6">G1</font>, <font color="Maroon" title="c7">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>assume </b><b>that </b><br/><a NAME="E1:47"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F2</font>
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c6">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">s</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font>;<br/><b>let </b><font color="Maroon" title="c9">t</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font>;<br/>



<b>assume </b><a NAME="E3:47"/><i><font color="Green" title="E41">A3</font></i>: 
(  <a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c8">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c9">t</font> &amp;  <a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c8">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c9">t</font> )
 ;<br/>

<a NAME="E4:47"/>
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"><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">B</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c3">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="cat_2.html#T33" title="CAT_2:th.33">CAT_2:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">S</font> = <font color="Maroon" title="c8">s</font>, <font color="Maroon" title="c11">T</font> = <font color="Maroon" title="c9">t</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">B</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c3">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<i><font color="Green" title="E42">A4</font></i>: <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">B</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">S</font> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">pr1</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">S</font>
<b>by </b><i><a class="ref" href="cat_2.html#D10" title="CAT_2:def.10">CAT_2:def 10</a></i>
<br/>.= 
 <a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c8">s</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:47"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="isocat_2.html#T35" target="_self" title="ISOCAT_2:th.35">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">pr1</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">T</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:47"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="isocat_2.html#E2:47"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="isocat_2.html#T35" target="_self" title="ISOCAT_2:th.35">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">B</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">T</font>
<b>by </b><i><a class="ref" href="cat_2.html#D10" title="CAT_2:def.10">CAT_2:def 10</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">B</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">S</font> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">pr2</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">S</font>
<b>by </b><i><a class="ref" href="cat_2.html#D11" title="CAT_2:def.11">CAT_2:def 11</a></i>
<br/>.= 
 <a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c8">s</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:47"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="isocat_2.html#T35" target="_self" title="ISOCAT_2:th.35">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">pr2</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">T</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:47"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="isocat_2.html#E2:47"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="isocat_2.html#T35" target="_self" title="ISOCAT_2:th.35">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">B</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">T</font>
<b>by </b><i><a class="ref" href="cat_2.html#D11" title="CAT_2:def.11">CAT_2:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E8:47"/>
<font color="Maroon" title="c8">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">t</font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E6:47"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="isocat_2.html#T5" target="_self" title="ISOCAT_2:th.5">Th5</a></i>;<br/>


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