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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> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/>

<b>given </b><font color="Maroon" title="c4">F1</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="c2">B</font>, <font color="Maroon" title="c5">G1</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="c2">B</font>,<font color="Maroon" title="c1">A</font><b> such that </b><a NAME="E1:20"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</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="E2:20"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">G1</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> 
 ; <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><br/>

<b>given </b><font color="Maroon" title="c6">F2</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="c2">B</font>,<font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c7">G2</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="c2">B</font><b> such that </b><a NAME="E3:20"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c6">F2</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>and </b><br/><a NAME="E4:20"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c6">F2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c7">G2</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><br/>

<b>take </b><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c6">F2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>; <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><br/>

<b>take </b><font color="Maroon" title="c9">G</font> = <font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c7">G2</font>;<br/>

<a NAME="E5:20"/>
 <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="c4">F1</font>,the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c4">F1</font> #) <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="c4">F1</font>,the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c4">F1</font> #)
 
;<br/>
<a NAME="E6:20"/><b>then </b><i><font color="Green" title="E9">A5</font></i>: 
<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">G1</font>,the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">G1</font> #) <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>
 
<b>by </b><i><a class="ref" href="yellow18.html#T3" target="_self" title="YELLOW18:th.3">Th3</a></i>;<br/>
<a NAME="E7:20"/>
 <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="c7">G2</font>,the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c7">G2</font> #) <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="c7">G2</font>,the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c7">G2</font> #)
 
;<br/>
<a NAME="E8:20"/><b>then </b><i><font color="Green" title="E10">A6</font></i>: 
<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">F2</font>,the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c6">F2</font> #) <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c7">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c7">G2</font>
 
<b>by </b><i><a class="ref" href="yellow18.html#T3" target="_self" title="YELLOW18:th.3">Th3</a></i>;<br/>
<a NAME="E9:20"/><i><font color="Green" title="E11">A7</font></i>: 
( <font color="Maroon" title="c5">G1</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="c2">B</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">G1</font>,the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">G1</font> #) &amp; <font color="Maroon" title="c6">F2</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="c2">B</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">F2</font>,the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c6">F2</font> #) )
 
<b>by </b><i><a class="ref" href="functor3.html#T5" title="FUNCTOR3:th.5">FUNCTOR3:5</a></i>;<br/>
<a NAME="E10:20"/><i><font color="Green" title="E12">A8</font></i>: 
( <font color="Maroon" title="c9">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c6">F2</font></span>)</span> &amp; <font color="Maroon" title="c8">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c8">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">G1</font></span>)</span> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c7">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c9">G</font> )
 
<b>by </b><i><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></i>;<br/>
<a NAME="E11:20"/><b>then </b>
( <font color="Maroon" title="c9">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c6">F2</font>,<font color="Maroon" title="c5">G1</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="c2">B</font></span>)</span> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">F2</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="c2">B</font></span>)</span> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:20"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="yellow18.html#E3:20"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="functor3.html#T35" title="FUNCTOR3:th.35">FUNCTOR3:35</a></i>;<br/>
<a NAME="E12:20"/><b>then </b>
( <font color="Maroon" title="c9">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c6">F2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c7">G2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:20"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="yellow18.html#E8:20"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="yellow18.html#E9:20"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="yellow18.html#E10:20"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="functor3.html#T36" title="FUNCTOR3:th.36">FUNCTOR3:36</a></i>;<br/>
<b>hence </b><a NAME="E13:20"/>
( <font color="Maroon" title="c9">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c8">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>  &amp; <font color="Maroon" title="c8">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c9">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>  )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:20"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="yellow18.html#E4:20"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="functor3.html#T33" title="FUNCTOR3:th.33">FUNCTOR3:33</a></i>;<br/>


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