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

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/>

<b>set </b><font color="Maroon" title="c2">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="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c3">C</font> =  <a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c1">A</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="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<b>thus </b><a NAME="E1:27"/>
for <font color="Olive" title="b1">x</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c1">A</font>,<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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> holds  <font color="Olive" title="b1">x</font> is   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D5" target="_self" title="CAT_5:def.5">CAT_5:def 5</a>,<a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">CAT_5:def 6</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c1">A</font>,<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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">D</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font>
 ;<br/><a NAME="E2:27_1"/><b>then </b>
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:27_1"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</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">D</font>,<font color="Maroon" title="c5">D</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c5">D</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c4">m</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c1">A</font>,<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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">C1</font>, <font color="Maroon" title="c6">C2</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>assume </b><a NAME="E1:27_2"/>
( <font color="Maroon" title="c5">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c6">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">m</font> )
 ;<br/><a NAME="E2:27_2"/><b>then </b><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c5">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c6">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">G</font> =  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">C2</font> ;<br/><b>take </b><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c7">G</font>;<br/><b>thus </b><a NAME="E4:27_2"/>
<font color="Maroon" title="c4">m</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">C1</font>,<font color="Maroon" title="c6">C2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">G</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E2:27_2"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">m1</font>, <font color="Maroon" title="c5">m2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c1">A</font>,<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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">A1</font>, <font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c8">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c9">F1</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c6">A1</font>,<font color="Maroon" title="c7">B</font>;<br/><b>let </b><font color="Maroon" title="c10">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font>;<br/>







<b>assume </b><a NAME="E4:27"/>
( <font color="Maroon" title="c4">m1</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">A1</font>,<font color="Maroon" title="c7">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">m2</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">B</font>,<font color="Maroon" title="c8">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/>

<a NAME="E5:27"/><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"><font color="Maroon" title="c6">A1</font>,<font color="Maroon" title="c7">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">F1</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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></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"><font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c10">F2</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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:27"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">A1</font>,<font color="Maroon" title="c7">B</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="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</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="c7">B</font>,<font color="Maroon" title="c8">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">F2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E7:27"/><b>then </b>
( <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">F2</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c9">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:27"/>
<font color="Maroon" title="c5">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">m1</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">A1</font>,<font color="Maroon" title="c8">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c9">F1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div>
