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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="E1:37"/><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/>

<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_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>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<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="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">B</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> = 
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">B</font>,<font color="Maroon">A</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">B</font>,<font color="Maroon">A</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">B</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="functor0.html#T30">FUNCTOR0:30</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><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>assume </b><a NAME="E1:37_2"/><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/><a NAME="E2:37_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">B</font>,<font color="Maroon">A</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">B</font>,<font color="Maroon">A</font></span>)</span> <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="functor0.html#D20">FUNCTOR0:def 20</a></i>;<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><span class="p1">(<span class="default"><span class="p2">(<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="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">B</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font> = 
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">B</font>,<font color="Maroon">A</font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="functor3.html#T7">FUNCTOR3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">B</font>,<font color="Maroon">A</font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">f</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">B</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="functor0.html#T32">FUNCTOR0:32</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:37"/>
<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="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">B</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K12">id</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/>


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