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

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

<b>set </b><font color="Maroon">F</font> =  <a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font>;<br/>
<a NAME="E1:12"/>
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  st  <a href="cat_1.html#K6">Hom</a> <font color="Olive">o1</font>,<font color="Olive">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">o1</font>,<font color="Olive">o2</font>  st <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">o2</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E17</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>, <font color="Maroon">x2</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/>

<b>assume </b><a NAME="E2:12_1"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E3:12_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E4:12_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">dd</font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E6:12_1"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<a NAME="E7:12_1"/><b>then </b>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x1</font></span>)</span>,<font color="Maroon">o2</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E8:12_1"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E9:12_1"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<a NAME="E10:12_1"/><b>then </b>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">o2</font>
 
<b>by </b><i><a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E11:12_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E12:12_1"/><i><font color="Green">E25</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T56">CAT_1:56</a></i>;<br/>
<a NAME="E13:12_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">dd</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>;<br/>
<b>then </b><i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x1</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">dd</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="ens_1.html#D24">ENS_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="cat_1.html#T59">CAT_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span> <a href="cat_1.html#K9">*</a> <font color="Maroon">x1</font>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T57">CAT_1:57</a></i>
;<br/>
<i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x2</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">dd</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">x2</font>
<b>by </b><i>, <a class="txt" href="yoneda_1.html#E1:12_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="ens_1.html#D24">ENS_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="cat_1.html#T59">CAT_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span> <a href="cat_1.html#K9">*</a> <font color="Maroon">x2</font>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T57">CAT_1:57</a></i>
;<br/>
<i><font color="Green">E42</font></i>:  <a href="cat_1.html#K3">cod</a> <font color="Maroon">x1</font> = 
<font color="Maroon">o2</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>
;<br/>
<i><font color="Green">E43</font></i>:  <a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font> = 
<font color="Maroon">o1</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>
;<br/>
<a NAME="E18:12_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x1</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">x1</font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:12_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x1</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">x1</font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></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"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x2</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">x2</font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E20:12_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x1</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">x1</font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></span><a href="mcart_1.html#K3">]</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"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x2</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">x2</font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E21:12_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x1</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">x1</font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x2</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">x2</font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E22:12_1"/><b>then </b>
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">x1</font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">x2</font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>
<a NAME="E23:12_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">x2</font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, <a class="txt" href="yoneda_1.html#E3:12_1"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<a NAME="E24:12_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></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"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E25:12_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></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"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E26:12_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></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"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E27:12_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></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"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E28:12_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></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"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E29:12_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E30:12_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o2</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E31:12_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:12_1"><i><font color="Green">E18</font></i></a>, , <a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12"/>
 <a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font> is <a href="yoneda_1.html#V1" target="_self">faithful</a>
 <b>by </b><i/>;<br/>


</div>
