<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c4">D</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  &amp; <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  &amp; <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a>  implies <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font> <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a> 
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a>  or <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font> <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c5">F</font> being   <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c6">G</font> being   <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">A</font><span class="kw"> such that </span><a NAME="E4:47"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 <span class="kw">and </span><br/><a NAME="E5:47"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c6">G</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c3">C</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow18.html#D2" target="_self" onmouseover="rs('yellow18/D2')" onmouseout="rh()">YELLOW18:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font> <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a> </span><br/>

<span class="kw">take </span><font color="Maroon" title="c7">dF</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow18.html#D2" target="_self" onmouseover="rs('yellow18/D2')" onmouseout="rh()">YELLOW18:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">B</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c7">dF</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c2">B</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c7">dF</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Olive" title="b1">G</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c4">D</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c8">dG</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">dG</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c7">dF</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c2">B</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c7">dF</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c8">dG</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c4">D</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )</span><br/>

<a NAME="E6:47"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c6">G</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c6">G</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor3.html#T5" onmouseover="rs('functor3/T5')" onmouseout="rh()">FUNCTOR3:5</a></span>;<br/>
<a NAME="E7:47"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor3.html#T5" onmouseover="rs('functor3/T5')" onmouseout="rh()">FUNCTOR3:5</a></span>;<br/>
<a NAME="E8:47"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C</font>)</span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="yellow18.html#T14" target="_self" onmouseover="rs('yellow18/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="lab"><font color="Green" title="E29">A8</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">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" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="yellow18.html#T3" target="_self" onmouseover="rs('yellow18/T3')" onmouseout="rh()">Th3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">dG</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
<br/>.= 
<font color="Maroon" title="c8">dG</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
<br/>.= 
<font color="Maroon" title="c8">dG</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c7">dF</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
;<br/>
<a NAME="E10:47"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="yellow18.html#T14" target="_self" onmouseover="rs('yellow18/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:47"/>
<font color="Maroon" title="c8">dG</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c7">dF</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c2">B</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E9:47"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="yellow18.html#T24" target="_self" onmouseover="rs('yellow18/T24')" onmouseout="rh()">Th24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">dF</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c8">dG</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c4">D</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> </span><br/>

<a NAME="E12:47"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">F</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor3.html#T5" onmouseover="rs('functor3/T5')" onmouseout="rh()">FUNCTOR3:5</a></span>;<br/>
<a NAME="E13:47"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor3.html#T5" onmouseover="rs('functor3/T5')" onmouseout="rh()">FUNCTOR3:5</a></span>;<br/>
<a NAME="E14:47"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
 <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>)</span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="yellow18.html#T14" target="_self" onmouseover="rs('yellow18/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="lab"><font color="Green" title="E33">A12</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c6">G</font></span>)</span></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">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" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:47"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="yellow18.html#T3" target="_self" onmouseover="rs('yellow18/T3')" onmouseout="rh()">Th3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">dF</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:47"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
<br/>.= 
<font color="Maroon" title="c7">dF</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor3.html#K4" title="FUNCTOR3:func.4">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
<br/>.= 
<font color="Maroon" title="c7">dF</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c8">dG</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T32" onmouseover="rs('functor0/T32')" onmouseout="rh()">FUNCTOR0:32</a></span>
;<br/>
<a NAME="E16:47"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c4">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E13:47"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="yellow18.html#T14" target="_self" onmouseover="rs('yellow18/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:47"/>
<font color="Maroon" title="c7">dF</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c8">dG</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c4">D</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E15:47"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="yellow18.html#T24" target="_self" onmouseover="rs('yellow18/T24')" onmouseout="rh()">Th24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
