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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font> is  <a href="yoneda_1.html#V1" title="YONEDA_1:attr.1">faithful</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">F</font> =  <a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font>;<br/>
<a NAME="E1:13"/>
for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font>  st <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x2</font> holds <br/><font color="Olive" title="b3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>  st <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:13_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>  st <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x2</font> implies <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font> )</span><br/>

<a NAME="E2:13_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:13_1"/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font></span><br/>

<a NAME="E4:13_1"/><span class="kw">then </span>
<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"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">x1</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">x1</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c5">x1</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></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">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yoneda_1.html#D4" target="_self" title="YONEDA_1:def.4">Def4</a></span>;<br/>
<a NAME="E5:13_1"/><span class="kw">then </span>
<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"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">x1</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">x1</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c5">x1</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></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"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c6">x2</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yoneda_1.html#D4" target="_self" title="YONEDA_1:def.4">Def4</a></span>;<br/>
<a NAME="E6:13_1"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">x1</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">x1</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c5">x1</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</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"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c6">x2</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></span>;<br/>
<a NAME="E7:13_1"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">x1</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">x1</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c5">x1</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c6">x2</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></span>;<br/>
<a NAME="E8:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c5">x1</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c6">x2</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></span>;<br/>
<a NAME="E9:13_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E13">A5</font></span>:  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">x1</font> = 
<font color="Maroon" title="c3">o1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>
;<br/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">x1</font> = 
<font color="Maroon" title="c4">o2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:13_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>
;<br/>
<a NAME="E12:13_1"/><span class="kw">then </span>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c5">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></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="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c6">x2</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:13_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:13_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="yoneda_1.html#D3" target="_self" title="YONEDA_1:def.3">Def3</a></span>;<br/>
<a NAME="E13:13_1"/><span class="kw">then </span>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c5">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c6">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yoneda_1.html#D3" target="_self" title="YONEDA_1:def.3">Def3</a></span>;<br/>
<a NAME="E14:13_1"/><span class="kw">then </span>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c5">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c6">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:13_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E15:13_1"/><span class="kw">then </span>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c5">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c6">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:13_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E16:13_1"/><span class="kw">then </span>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c5">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c6">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E17:13_1"/><span class="kw">then </span>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c5">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c6">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E18:13_1"/><span class="kw">then </span>
<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"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c5">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></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="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c6">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></span>;<br/>
<a NAME="E19:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c5">x1</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c6">x2</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">dd</font> =  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E21:13_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></span>;<br/>
<a NAME="E22:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c7">dd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></span>;<br/>
<a NAME="E23:13_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></span>;<br/>
<a NAME="E24:13_1"/><span class="kw">then </span>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<font color="Maroon" title="c4">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E25:13_1"/><span class="kw">then </span>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c6">x2</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">dd</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:13_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="ens_1.html#D24" title="ENS_1:def.24">ENS_1:def 24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T59" title="CAT_1:th.59">CAT_1:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E21:13_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="cat_1.html#T57" title="CAT_1:th.57">CAT_1:57</a></span>
;<br/>
<a NAME="E27:13_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c4">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></span>;<br/>
<a NAME="E28:13_1"/><span class="kw">then </span>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<font color="Maroon" title="c4">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:13_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E29:13_1"/><span class="kw">then </span>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c5">x1</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">dd</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:13_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="ens_1.html#D24" title="ENS_1:def.24">ENS_1:def 24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T59" title="CAT_1:th.59">CAT_1:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">o2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E21:13_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c5">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="cat_1.html#T57" title="CAT_1:th.57">CAT_1:57</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E31:13_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:13_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E19:13_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:13"/>
 <a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font> is  <a href="yoneda_1.html#V1" title="YONEDA_1:attr.1">faithful</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yoneda_1.html#D6" target="_self" title="YONEDA_1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
