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<b>let </b><font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">A</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>assume </b><a NAME="E1:16"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">F</font> <a href="isocat_1.html#NR1" target="_self">is_an_isomorphism</a> 
 ;<br/>

<a NAME="E2:16"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font> ;<br/>
<a NAME="E4:16"/><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</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="E5:16"/><i><font color="Green">E23</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">F</font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="isocat_1.html#K1" target="_self">"</a> </span>)</span> = 
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="cat_1.html#K15">id</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="txt" href="isocat_1.html#E1:16"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="isocat_1.html#K1" target="_self">"</a> </span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="isocat_1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
;<br/>


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