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<b>let </b><font color="Maroon" title="c1">B</font>, <font color="Maroon" title="c2">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">B</font>;<br/>





<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="isocat_1.html#NR1" title="ISOCAT_1:NR.1">is_an_isomorphism</a> 
 ;<br/>

<a NAME="E2:16"/><b>then </b><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="isocat_1.html#D3" target="_self" title="ISOCAT_1:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">A</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">B</font> ;<br/>
<a NAME="E4:16"/><i><font color="Green" title="E15">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:16"/><i><font color="Green" title="E16">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">B</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E1:16"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="isocat_1.html#D3" target="_self" title="ISOCAT_1:def.3">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:16"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="isocat_1.html#D2" target="_self" title="ISOCAT_1:def.2">Def2</a></i>
<br/>.= 
 <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">B</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E2:16"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="isocat_1.html#E5:16"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:16"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="isocat_1.html#D2" target="_self" title="ISOCAT_1:def.2">Def2</a></i>
<br/>.= 
 <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E2:16"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="isocat_1.html#E4:16"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a></i>
;<br/>


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