<?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="E1:13"/><i><font color="Green">E17</font></i>: 
 <a href="cat_1.html#K12">Obj</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_1"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</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="13_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">x1</font>, <font color="Maroon">c1</font> = <font color="Maroon">x2</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:13_1_1"/>
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d</font> = <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> as   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E5:13_1_1"/>
<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> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">X</font> =  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</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><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span>;<br/>
<i><font color="Green">E21</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</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><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> = 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E2:13_1_1_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</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><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> holds <br/><span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</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><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</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><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">y</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:13_1_1_1_2"/>
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E4:13_1_1_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E5:13_1_1_1_2"/><i><font color="Green">E23</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">z</font>,<font color="Maroon">z</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/>
<b>reconsider </b><font color="Maroon">zz</font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">z</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">z</font>,<font color="Maroon">z</font> ;<br/>
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</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><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">z</font> = 
<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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span>,<font color="Maroon">z</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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span>,<font color="Maroon">z</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> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">z</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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span>,<font color="Maroon">z</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> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">z</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> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T53">ENS_1:53</a></i>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#D22">ENS_1:def 22</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">zz</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T15">NATTRA_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="nattra_1.html#T21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E8:13_1_1_1_2"/>
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</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><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:13_1_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</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><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_2.html#R2">=</a>  <a href="nattra_1.html#K7">id</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<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> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> = 
<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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></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"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></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/>.= 
<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"><font color="Maroon">c</font></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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></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"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></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="cat_1.html#T44">CAT_1:44</a></i>
<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"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></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"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></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#T44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E5:13_1_1_1"/>
<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> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:13_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T31">OPPCAT_1:31</a></i>;<br/>
<a NAME="E7:13_1_1"/>
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d1</font> = <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> as   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E9:13_1_1"/>
<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> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">d1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">X</font> =  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span>;<br/>
<i><font color="Green">E26</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> = 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E2:13_1_1_2"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> holds <br/><span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">y</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:13_1_1_2_2"/>
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E4:13_1_1_2_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E5:13_1_1_2_2"/><i><font color="Green">E41</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">z</font>,<font color="Maroon">z</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/>
<b>reconsider </b><font color="Maroon">zz</font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">z</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">z</font>,<font color="Maroon">z</font> ;<br/>
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">z</font> = 
<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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span>)</span>,<font color="Maroon">z</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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span>)</span>,<font color="Maroon">z</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> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<font color="Maroon">z</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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span>)</span>,<font color="Maroon">z</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> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<font color="Maroon">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<font color="Maroon">z</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> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<font color="Maroon">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T53">ENS_1:53</a></i>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<font color="Maroon">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#D22">ENS_1:def 22</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">zz</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T15">NATTRA_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="nattra_1.html#T21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E8:13_1_1_2_2"/>
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:13_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_2.html#R2">=</a>  <a href="nattra_1.html#K7">id</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<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> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span> = 
<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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></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"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></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/>.= 
<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"><font color="Maroon">c1</font></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> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></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"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span></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="cat_1.html#T44">CAT_1:44</a></i>
<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"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></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"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c1</font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">d1</font>
<b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E5:13_1_1_2"/>
<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> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">d1</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:13_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T31">OPPCAT_1:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c1</font>,<font color="Maroon">c1</font> ;<br/>
<a NAME="E12:13_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<font color="Maroon">c1</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="E13:13_1_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">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K20">hom?-</a> <font color="Maroon">c1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, <a class="ref" href="ens_1.html#D22">ENS_1:def 22</a></i>;<br/>
<a NAME="E14:13_1_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">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c</font>,<font color="Maroon">f</font></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">c1</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c1</font>,<font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="ens_1.html#D22">ENS_1:def 22</a></i>;<br/>
<a NAME="E15:13_1_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">c</font>,<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#K6">Hom</a> <font color="Maroon">c</font>,<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="ens_1.html#K18">hom</a> <font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</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">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c1</font>,<font color="Maroon">f</font></span>)</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="E16:13_1_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">c</font>,<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#K6">Hom</a> <font color="Maroon">c</font>,<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="ens_1.html#K18">hom</a> <font color="Maroon">c</font>,<font color="Maroon">f</font></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">c1</font>,<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#K6">Hom</a> <font color="Maroon">c1</font>,<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="ens_1.html#K18">hom</a> <font color="Maroon">c1</font>,<font color="Maroon">f</font></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="E17:13_1_1"/><b>then </b>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>
<a NAME="E18:13_1_1"/><b>then </b>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<font color="Maroon">c1</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E19:13_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<font color="Maroon">c1</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E20:13_1_1"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c1</font>,<font color="Maroon">c1</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">c1</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">c1</font> <b>by </b><i>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E22:13_1_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, , <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E23:13_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_1"/>
 <a href="cat_1.html#K12">Obj</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:13"/>
 <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/>
<b>hence </b><a NAME="E3:13"/>
 <a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">A</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, </i>;<br/>


</div>
