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<b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">B</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">F</font> be  <a href="functor0.html#V16">contravariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>



<b>assume </b><a NAME="E1:53"/>
( <font color="Maroon">A</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">B</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow20.html#D5" target="_self">YELLOW20:def 5</a><br/>

<b>given </b><font color="Maroon">G</font> being  <a href="functor0.html#V16">contravariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><a NAME="E2:53"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">G</font> is <a href="functor0.html#V21">bijective</a>
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">a'</font>, <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st <font color="Olive">a'</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> holds <br/><font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Olive">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font>
 <b>and </b><br/><a NAME="E4:53"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b'</font>, <font color="Olive">c'</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b'</font>,<font color="Olive">c'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">c'</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> holds <br/>for <font color="Olive">f'</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b'</font>,<font color="Olive">c'</font><br/> for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b</font>,<font color="Olive">c</font>  st <font color="Olive">f'</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> holds <br/><font color="Maroon">G</font> <a href="functor0.html#K8">.</a> <font color="Olive">f'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F</font>,<font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font>
 ;<br/>

<a NAME="E5:53"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> holds  <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font>
 
<b>by </b><i><a class="txt" href="yellow20.html#E4:53"><i><font color="Green">E33</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><a NAME="E2:53_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></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="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="functor0.html#D20">FUNCTOR0:def 20</a></i>;<br/><b>hence </b><font color="Maroon">F</font> <a href="functor0.html#K8">.</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="functor0.html#K8">.</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="yellow20.html#E5:53"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow20.html#T4" target="_self">Th4</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E7:53"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F</font> #) <a href="hidden.html#R1">=</a>  <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">G</font> #)
 
<b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a>, <a class="ref" href="yellow18.html#T2">YELLOW18:2</a></i>;<br/>
<a NAME="E8:53"/>
( <font color="Maroon">G</font> is <a href="functor0.html#V19">injective</a> &amp; <font color="Maroon">G</font> is <a href="functor0.html#V20">surjective</a> )
 
<b>by </b><i><a class="txt" href="yellow20.html#E3:53"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E9:53"/><b>then </b>
( <font color="Maroon">G</font> is <a href="functor0.html#V4">one-to-one</a> &amp; <font color="Maroon">G</font> is <a href="functor0.html#V17">faithful</a> &amp; <font color="Maroon">G</font> is <a href="functor0.html#V18">full</a> &amp; <font color="Maroon">G</font> is <a href="functor0.html#V5">onto</a> )
 
<b>by </b><i><a class="ref" href="functor0.html#D34">FUNCTOR0:def 34</a>, <a class="ref" href="functor0.html#D35">FUNCTOR0:def 35</a></i>;<br/>
<b>hence </b><a NAME="E10:53"/>
( the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a> &amp;  ex <font color="Olive">f</font> being   <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> st <br/>( <font color="Olive">f</font> <a href="pboole.html#R6">=</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F</font> &amp; <font color="Olive">f</font> is <a href="msualg_3.html#V2">"onto"</a> ) &amp; the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> is <a href="funct_2.html#V2">onto</a> )
 <b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="functor0.html#D7">FUNCTOR0:def 7</a>, <a class="ref" href="functor0.html#D8">FUNCTOR0:def 8</a>, <a class="ref" href="functor0.html#D31">FUNCTOR0:def 31</a>, <a class="ref" href="functor0.html#D33">FUNCTOR0:def 33</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D7">FUNCTOR0:def 7</a>,<a class="ref" href="functor0.html#D8">FUNCTOR0:def 8</a>,<a class="ref" href="functor0.html#D31">FUNCTOR0:def 31</a>,<a class="ref" href="functor0.html#D33">FUNCTOR0:def 33</a>,<a class="ref" href="functor0.html#D34">FUNCTOR0:def 34</a>,<a class="ref" href="functor0.html#D35">FUNCTOR0:def 35</a>,<a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a><br/>


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