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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>, <font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">D</font> be   <a href="altcat_1.html#NM4">category</a>;<br/>

<b>assume </b><a NAME="E1:50"/><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>  &amp; <font color="Maroon">C</font>,<font color="Maroon">D</font> <a href="yellow18.html#R2" target="_self">are_opposite</a>  )
 ;<br/>

<b>given </b><font color="Maroon">F</font> being  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">C</font>, <font color="Maroon">G</font> being  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font><b> such that </b><a NAME="E2:50"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">G</font> <a href="functor3.html#K1">*</a> <font color="Maroon">F</font>, <a href="functor0.html#K12">id</a> <font color="Maroon">A</font> <a href="functor3.html#R1">are_naturally_equivalent</a> 
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">F</font> <a href="functor3.html#K1">*</a> <font color="Maroon">G</font>, <a href="functor0.html#K12">id</a> <font color="Maroon">C</font> <a href="functor3.html#R1">are_naturally_equivalent</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D2" target="_self">YELLOW18:def 2</a><br/>

<b>take </b><font color="Maroon">dF</font> = <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">C</font>,<font color="Maroon">D</font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">F</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D2" target="_self">YELLOW18:def 2</a><br/>

<b>take </b><font color="Maroon">dG</font> = <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#K3">*</a> <font color="Maroon">G</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">D</font>,<font color="Maroon">C</font></span>)</span>;<br/>

<a NAME="E4:50"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">G</font> <a href="functor3.html#K1">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">C</font></span>)</span> <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> #) &amp; <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#K3">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="functor3.html#T5">FUNCTOR3:5</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green">E43</font></i>: 
 <a href="functor0.html#K12">id</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">D</font>,<font color="Maroon">C</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">C</font>,<font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E48</font></i>: <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#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K1">*</a> <font color="Maroon">F</font></span>)</span></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> = 
<span class="p1">(<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">A</font>,<font color="Maroon">B</font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">F</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>
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
<br/>.= 
<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#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor3.html#K4">*</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>
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
<br/>.= 
<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#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K1">*</a> <span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor3.html#K4">*</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>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<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">A</font>,<font color="Maroon">B</font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="functor3.html#K3">*</a> <span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor3.html#K4">*</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>
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">dG</font> <a href="functor3.html#K4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">C</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor3.html#K4">*</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>
<b>by </b><i>, <a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
<br/>.= 
<font color="Maroon">dG</font> <a href="functor3.html#K1">*</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">C</font>,<font color="Maroon">D</font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor3.html#K4">*</a> <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></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
<br/>.= 
<font color="Maroon">dG</font> <a href="functor3.html#K1">*</a> <font color="Maroon">dF</font>
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
;<br/>
<a NAME="E7:50"/>
<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#K3">*</a> <span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">A</font></span>)</span></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/>
<b>hence </b><a NAME="E8:50"/>
<font color="Maroon">dG</font> <a href="functor3.html#K1">*</a> <font color="Maroon">dF</font>, <a href="functor0.html#K12">id</a> <font color="Maroon">B</font> <a href="functor3.html#R1">are_naturally_equivalent</a> 
 <b>by </b><i>, , , </i>;<br/>

<a NAME="E9:50"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">F</font> <a href="functor3.html#K1">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">A</font></span>)</span> <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">F</font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F</font> #) &amp; <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">C</font>,<font color="Maroon">D</font></span>)</span> <a href="functor3.html#K3">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">C</font>,<font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="functor3.html#T5">FUNCTOR3:5</a></i>;<br/>
<a NAME="E10:50"/><i><font color="Green">E50</font></i>: 
 <a href="functor0.html#K12">id</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</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="functor3.html#K2">*</a> <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>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E53</font></i>: <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">C</font>,<font color="Maroon">D</font></span>)</span> <a href="functor3.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor3.html#K1">*</a> <font color="Maroon">G</font></span>)</span></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">D</font>,<font color="Maroon">C</font></span>)</span> = 
<span class="p1">(<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">C</font>,<font color="Maroon">D</font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">G</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">D</font>,<font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
<br/>.= 
<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">C</font>,<font color="Maroon">D</font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">D</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
<br/>.= 
<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">C</font>,<font color="Maroon">D</font></span>)</span> <a href="functor3.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor3.html#K1">*</a> <span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">D</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<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">C</font>,<font color="Maroon">D</font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor3.html#K3">*</a> <span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">D</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">dF</font> <a href="functor3.html#K4">*</a> <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></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">D</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
<br/>.= 
<font color="Maroon">dF</font> <a href="functor3.html#K1">*</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">A</font>,<font color="Maroon">B</font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K4">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">D</font>,<font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
<br/>.= 
<font color="Maroon">dF</font> <a href="functor3.html#K1">*</a> <font color="Maroon">dG</font>
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>
;<br/>
<a NAME="E12:50"/>
<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">C</font>,<font color="Maroon">D</font></span>)</span> <a href="functor3.html#K3">*</a> <span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">C</font></span>)</span></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">D</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K12">id</a> <font color="Maroon">D</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E13:50"/>
<font color="Maroon">dF</font> <a href="functor3.html#K1">*</a> <font color="Maroon">dG</font>, <a href="functor0.html#K12">id</a> <font color="Maroon">D</font> <a href="functor3.html#R1">are_naturally_equivalent</a> 
 <b>by </b><i>, , , </i>;<br/>


</div>
