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<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E1:63"/>
<font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:63"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#T31">NATTRA_1:31</a></i>;<br/>
<a NAME="E3:63"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D11">NATTRA_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of  <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font>,<font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font><b> such that </b><br/><a NAME="E5:63"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">s</font> is <a href="nattra_1.html#V1">invertible</a>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D11">NATTRA_1:def 11</a></i>;<br/>
<b>thus </b><a NAME="E6:63"/>
<font color="Maroon">F</font> is <a href="cat_1.html#V10">faithful</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">a'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D24">CAT_1:def 24</a><br/>

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E23</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:63_1"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span> <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#T126">CAT_1:126</a></i>;<br/>
<a NAME="E3:63_1"/><i><font color="Green">E29</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">f1</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">a'</font>, <font color="Maroon">f2</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">a'</font>;<br/>

<b>assume </b><a NAME="E4:63_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font>
 ;<br/>

<i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
;<br/>
<a NAME="E6:63_1"/>
<font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font> is <a href="cat_1.html#V5">invertible</a>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#D10">NATTRA_1:def 10</a></i>;<br/>
<a NAME="E7:63_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font> is <a href="cat_1.html#V3">monic</a>
 
<b>by </b><i><a class="ref" href="isocat_1.html#T11" target="_self">Th11</a>, <a class="ref" href="cat_1.html#T73">CAT_1:73</a></i>;<br/>
<i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f1</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="isocat_1.html#T12" target="_self">Th12</a>, <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>
;<br/>
<b>thus </b><font color="Maroon">f1</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f1</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#T16">NATTRA_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2</font>
<b>by </b><i><a class="ref" href="isocat_1.html#T10" target="_self">Th10</a>, <a class="ref" href="isocat_1.html#T11" target="_self">Th11</a>, , , <a class="ref" href="cat_1.html#T60">CAT_1:60</a></i>
<br/>.= 
<font color="Maroon">f2</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#T16">NATTRA_1:16</a></i>
;<br/>


</div><b>end;</b></div>


</div>
