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<b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E25">A1</font></i>: 
( <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>given </b><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><b> such that </b><a NAME="E3:50"/><i><font color="Green" title="E26">A2</font></i>: 
<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#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c1">A</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 <b>and </b><br/><a NAME="E4:50"/><i><font color="Green" title="E27">A3</font></i>: 
<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#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c3">C</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D2" target="_self" title="YELLOW18:def.2">YELLOW18:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>take </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D2" target="_self" title="YELLOW18:def.2">YELLOW18:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K12" title="FUNCTOR0:func.12">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#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c4">D</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )</span><br/>

<b>take </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K12" title="FUNCTOR0:func.12">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#K12" title="FUNCTOR0:func.12">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="E5:50"/><i><font color="Green" title="E28">A4</font></i>: 
( <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#K12" title="FUNCTOR0:func.12">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> #) &amp; <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#K12" title="FUNCTOR0:func.12">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> )
 
<b>by </b><i><a class="ref" href="functor3.html#T5" title="FUNCTOR3:th.5">FUNCTOR3:5</a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green" title="E29">A5</font></i>: 
 <a href="functor0.html#K12" title="FUNCTOR0:func.12">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>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:50"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="yellow18.html#T15" target="_self" title="YELLOW18:th.15">Th15</a></i>;<br/>
<i><font color="Green" title="E30">A6</font></i>: <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>
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
<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>
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
<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#K12" title="FUNCTOR0:func.12">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>
<b>by </b><i><a class="txt" href="yellow18.html#E5:50"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="yellow18.html#T3" target="_self" title="YELLOW18:th.3">Th3</a></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" 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#K12" title="FUNCTOR0:func.12">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>
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
<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>
<b>by </b><i><a class="txt" href="yellow18.html#E6:50"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
<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>
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
<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>
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
;<br/>
<a NAME="E8:50"/>
<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#K12" title="FUNCTOR0:func.12">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#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:50"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="yellow18.html#E5:50"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="yellow18.html#T15" target="_self" title="YELLOW18:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E9:50"/>
<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#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c2">B</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 <b>by </b><i><a class="txt" href="yellow18.html#E1:50"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="yellow18.html#E3:50"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="yellow18.html#E7:50"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="yellow18.html#T25" target="_self" title="YELLOW18:th.25">Th25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K12" title="FUNCTOR0:func.12">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="E10:50"/><i><font color="Green" title="E31">A7</font></i>: 
( <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#K12" title="FUNCTOR0:func.12">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> #) &amp; <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#K12" title="FUNCTOR0:func.12">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> )
 
<b>by </b><i><a class="ref" href="functor3.html#T5" title="FUNCTOR3:th.5">FUNCTOR3:5</a></i>;<br/>
<a NAME="E11:50"/><i><font color="Green" title="E32">A8</font></i>: 
 <a href="functor0.html#K12" title="FUNCTOR0:func.12">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>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:50"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="yellow18.html#T15" target="_self" title="YELLOW18:th.15">Th15</a></i>;<br/>
<i><font color="Green" title="E33">A9</font></i>: <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>
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
<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>
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
<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#K12" title="FUNCTOR0:func.12">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>
<b>by </b><i><a class="txt" href="yellow18.html#E10:50"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="yellow18.html#T3" target="_self" title="YELLOW18:th.3">Th3</a></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" 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#K12" title="FUNCTOR0:func.12">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>
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
<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>
<b>by </b><i><a class="txt" href="yellow18.html#E11:50"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
<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>
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
<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>
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>
;<br/>
<a NAME="E13:50"/>
<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#K12" title="FUNCTOR0:func.12">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#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:50"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="yellow18.html#E10:50"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="yellow18.html#T15" target="_self" title="YELLOW18:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E14:50"/>
<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#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c4">D</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 <b>by </b><i><a class="txt" href="yellow18.html#E1:50"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="yellow18.html#E4:50"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="yellow18.html#E12:50"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="yellow18.html#T25" target="_self" title="YELLOW18:th.25">Th25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
