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

<b>let </b><font color="Maroon">C1</font> be  <a href="cat_1.html#V1">strict</a> <a href="cat_1.html#NM4">Category</a>, <font color="Maroon">C2</font> be  <a href="cat_1.html#V1">strict</a> <a href="cat_1.html#NM4">Category</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59_1_2"/><i><font color="Green">E194</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:59_1_2"/><i><font color="Green">E195</font></i>: 
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E3:59_1_2"/><i><font color="Green">E196</font></i>: 
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C1</font> holds <br/> (  <a href="cat_1.html#K2">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  )
 <b>and </b><br/><a NAME="E4:59_1_2"/><i><font color="Green">E197</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C1</font>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font>
 <b>and </b><br/><a NAME="E5:59_1_2"/><i><font color="Green">E198</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C1</font>  st <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font> holds <br/> ex <font color="Olive">F</font>, <font color="Olive">F1</font>, <font color="Olive">F2</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> ex <font color="Olive">t</font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F</font>,<font color="Olive">F1</font> ex <font color="Olive">t1</font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F1</font>,<font color="Olive">F2</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">F</font>,<font color="Olive">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">F1</font>,<font color="Olive">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">t1</font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">F</font>,<font color="Olive">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Olive">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E6:59_1_2"/><i><font color="Green">E199</font></i>: 
for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C1</font><br/> for <font color="Olive">F</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>  st <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> holds <br/> <a href="cat_1.html#K10">id</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">F</font>,<font color="Olive">F</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Olive">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:59_1_2"/><i><font color="Green">E200</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:59_1_2"/><i><font color="Green">E201</font></i>: 
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E9:59_1_2"/><i><font color="Green">E202</font></i>: 
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C2</font> holds <br/> (  <a href="cat_1.html#K2">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  )
 <b>and </b><br/><a NAME="E10:59_1_2"/><i><font color="Green">E203</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C2</font>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C2</font>
 <b>and </b><br/><a NAME="E11:59_1_2"/><i><font color="Green">E204</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C2</font>  st <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C2</font> holds <br/> ex <font color="Olive">F</font>, <font color="Olive">F1</font>, <font color="Olive">F2</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> ex <font color="Olive">t</font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F</font>,<font color="Olive">F1</font> ex <font color="Olive">t1</font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F1</font>,<font color="Olive">F2</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">F</font>,<font color="Olive">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">F1</font>,<font color="Olive">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">t1</font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">F</font>,<font color="Olive">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Olive">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E12:59_1_2"/><i><font color="Green">E205</font></i>: 
for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C2</font><br/> for <font color="Olive">F</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>  st <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> holds <br/> <a href="cat_1.html#K10">id</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">F</font>,<font color="Olive">F</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Olive">F</font></span>)</span></span><a href="tarski.html#K4">]</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>thus </b><a NAME="E1:59_1_2_1"/>
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C2</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E2:59_1_2_1"/>
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C1</font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_2_1_1"/>
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C2</font>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D8">FUNCT_2:def 8</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C1</font>;<br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C2</font> <b>by </b><i>, </i>;<br/>
<b>thus </b>the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">a</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>

;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:59_1_2_1"/>
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C1</font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_2_1_2"/>
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C2</font>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D8">FUNCT_2:def 8</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C1</font>;<br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C2</font> <b>by </b><i>, </i>;<br/>
<b>thus </b>the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">a</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>

;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_3"><b>now </b></a><div class="add"><a NAME="E1:59_1_2_1_3"/><i><font color="Green">E206</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C1</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>reconsider </b><font color="Maroon">S1</font> =  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font>, <font color="Maroon">S2</font> =  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C2</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C1</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i>, , <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C1</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C1</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><a NAME="E1:59_1_2_1_3_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">g1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C1</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">g2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C2</font> <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><i><font color="Green">E214</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_2_1_3_1_1"/>
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">f1</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 ;<br/><b>then consider </b><font color="Maroon">F</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F</font>,<font color="Maroon">F1</font>, <font color="Maroon">t1</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E3:59_1_2_1_3_1_1"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E4:59_1_2_1_3_1_1"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">f1</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/> <a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E6:59_1_2_1_3_1_1"/>
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">f2</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_2_1_3_1_2"/>
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">f2</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font>
 ;<br/><b>then consider </b><font color="Maroon">F</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F</font>,<font color="Maroon">F1</font>, <font color="Maroon">t1</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E3:59_1_2_1_3_1_2"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E4:59_1_2_1_3_1_2"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">f2</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/> <a href="cat_1.html#K2">dom</a> <font color="Maroon">g1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E6:59_1_2_1_3_1_2"/>
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">f1</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:59_1_2_1_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#T3" target="_self">Th3</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:59_1_2_1_3"/><i><font color="Green">E217</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E5:59_1_2_1_3"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">g</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C1</font> <b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E7:59_1_2_1_3"/><i><font color="Green">E219</font></i>: 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><b>then consider </b><font color="Maroon">F</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F</font>,<font color="Maroon">F1</font>, <font color="Maroon">t1</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E9:59_1_2_1_3"/><i><font color="Green">E220</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a></i>;<br/><b>consider </b><font color="Maroon">F'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F'</font>,<font color="Maroon">F1'</font>, <font color="Maroon">t1'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E11:59_1_2_1_3"/><i><font color="Green">E221</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1'</font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1'</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, , , , <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E12:59_1_2_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E13:59_1_2_1_3"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F'</font> &amp; <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F1'</font> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2'</font> &amp; <font color="Maroon">t</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t'</font> &amp; <font color="Maroon">t1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t1'</font> )
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E14:59_1_2_1_3"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="nattra_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:59_1_2_1"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>thus </b><a NAME="E6:59_1_2_1"/>
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C1</font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_2_1_4"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C2</font>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D8">FUNCT_2:def 8</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C1</font>;<br/>

<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">a</font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> <b>by </b><i>, <a class="ref" href="cat_2.html#D3">CAT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C2</font> <b>by </b><i>, </i>;<br/>
<b>thus </b>the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font>

<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>

;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:59_1_2"/>
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 <b>by </b><i>, </i>;<br/>


</div>
