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

<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="cat_1.html#NM4">Category</a>;<br/>



<b>reconsider </b><font color="Maroon">T</font> =  <a href="funct_3.html#K9">pr1</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11" target="_self">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="cat_2.html#K11" target="_self">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:56_1"/>
for <font color="Olive">cd</font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1"><a href="cat_2.html#K11" target="_self">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="cat_2.html#K11" target="_self">:]</a></span>  ex <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> st <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">cd</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">cd</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1"><a href="cat_2.html#K11" target="_self">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="cat_2.html#K11" target="_self">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">c</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">d</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E2:56_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">cd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="cat_2.html#K12" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E3:56_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">cd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span><a href="cat_2.html#K13" target="_self">]</a></span>
 
<b>by </b><i><a class="ref" href="cat_2.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E4:56_1_1"/>
<font color="Maroon">T</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:56_1_1"/>
 ex <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> st <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">cd</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">c</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:56_1"/>
for <font color="Olive">fg</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1"><a href="cat_2.html#K11" target="_self">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="cat_2.html#K11" target="_self">:]</a></span> holds <br/> ( <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">fg</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">fg</font></span>)</span></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">fg</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">fg</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">fg</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1"><a href="cat_2.html#K11" target="_self">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="cat_2.html#K11" target="_self">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E2:56_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E3:56_1_2"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span><a href="cat_2.html#K12" target="_self">]</a></span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span><a href="cat_2.html#K12" target="_self">]</a></span> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E2:56_1_2"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E4:56_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
(  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">fg</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="cat_2.html#K13" target="_self">]</a></span> &amp;  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">fg</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="cat_2.html#K13" target="_self">]</a></span> )
 
<b>by </b><i><a class="ref" href="cat_2.html#T4" target="_self">Th4</a>, <a class="txt" href="cat_2.html#E6:56_1_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E5:56_1_2"/>
<font color="Maroon">T</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg</font>
 
<b>by </b><i><a class="ref" href="cat_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:56_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">fg</font></span>)</span></span>)</span> = 
<font color="Maroon">T</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a>, </i>
;<br/>

<b>thus </b><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">fg</font></span>)</span></span>)</span> = 
<font color="Maroon">T</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a>, </i>
;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">fg</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1"><a href="cat_2.html#K11" target="_self">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="cat_2.html#K11" target="_self">:]</a></span>, <font color="Maroon">fg'</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1"><a href="cat_2.html#K11" target="_self">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="cat_2.html#K11" target="_self">:]</a></span>;<br/><b>assume </b><a NAME="E3:56_1"/><i><font color="Green">E25</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">fg'</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">fg</font>
 ;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E5:56_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><b>consider </b><font color="Maroon">f'</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">g'</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E7:56_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">fg'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f'</font>,<font color="Maroon">g'</font></span><a href="cat_2.html#K13" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E8:56_1"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f'</font>,<font color="Maroon">g'</font></span><a href="cat_2.html#K13" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f'</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g'</font></span>)</span></span><a href="cat_2.html#K12" target="_self">]</a></span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span><a href="cat_2.html#K12" target="_self">]</a></span> )
 <b>by </b><i/>;<br/><a NAME="E9:56_1"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="ref" href="cat_2.html#T4" target="_self">Th4</a>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E10:56_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">fg'</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span></span><a href="cat_2.html#K13" target="_self">]</a></span> &amp; <font color="Maroon">T</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f'</font>,<font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> &amp; <font color="Maroon">T</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/><a NAME="E11:56_1"/>
( <font color="Maroon">T</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f'</font>,<font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg'</font> &amp; <font color="Maroon">T</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E12:56_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/><b>thus </b><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">fg'</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">fg</font></span>)</span> = 
<font color="Maroon">T</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg'</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:56"/>
 <a href="funct_3.html#K9">pr1</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> is    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" target="_self">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="cat_2.html#K11" target="_self">:]</a></span>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T96">CAT_1:96</a></i>;<br/>


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