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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>assume </b><a NAME="E1:38"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="yellow18.html#R2" target="_self">are_opposite</a> 
 ;<br/>

<b>set </b><font color="Maroon">F</font> =  <a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>3</sub></font>;<br/>
<a NAME="E2:38"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> holds  <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:38"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</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">a</font>,<font color="Olive">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>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds  <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:38"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) holds <br/><font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 
;<br/>
<a NAME="E5:38"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</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">a</font>,<font color="Olive">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>  holds <br/>for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font>  st <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">g</font>) holds <br/><font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 
;<br/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_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">B</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/><a NAME="E2:38_1"/><i><font color="Green">E49</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#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b'</font>,<font color="Maroon">a'</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, </i>;<br/><b>assume </b><a NAME="E3:38_1"/><i><font color="Green">E50</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/><b>thus </b><a NAME="E4:38_1"/>
 ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> ex <font color="Olive">g</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">c</font>,<font color="Olive">d</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">c</font>) &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</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="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Olive">g</font>) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E7:38"/>
 <a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> is <a href="functor0.html#V21">bijective</a>
 
<b>from </b><i><a class="ref" href="yellow18.html#S12" target="_self">YELLOW18:sch 12</a>(, , , , );<br/></i>
<b>hence </b><a NAME="E8:38"/>
 <a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> is <a href="functor0.html#V21">bijective</a>
 ;<br/>


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