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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM4">category</a>;<br/>

<b>given </b><font color="Maroon">c<sub>4</sub></font> being  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> being  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:20"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>, <a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#R1">are_naturally_equivalent</a> 
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>, <a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>2</sub></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>given </b><font color="Maroon">c<sub>6</sub></font> being  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> being  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E3:20"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font>, <a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#R1">are_naturally_equivalent</a> 
 <b>and </b><br/><a NAME="E4:20"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>, <a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>3</sub></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">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>; <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">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>;<br/>

<a NAME="E5:20"/>
 <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>4</sub></font> #) <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">c<sub>4</sub></font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>4</sub></font> #)
 
;<br/>
<a NAME="E6:20"/><b>then </b><i><font color="Green">E9</font></i>: 
<a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>5</sub></font> #) <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="yellow18.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:20"/>
 <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>7</sub></font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>7</sub></font> #) <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">c<sub>7</sub></font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>7</sub></font> #)
 
;<br/>
<a NAME="E8:20"/><b>then </b><i><font color="Green">E10</font></i>: 
<a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>6</sub></font> #) <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="yellow18.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E9:20"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>5</sub></font> #) &amp; <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>6</sub></font> #) )
 
<b>by </b><i><a class="ref" href="functor3.html#T5">FUNCTOR3:5</a></i>;<br/>
<a NAME="E10:20"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>;<br/>
<a NAME="E11:20"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor3.html#R1">are_naturally_equivalent</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor3.html#R1">are_naturally_equivalent</a>  )
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:20"><i><font color="Green">E6</font></i></a>, <a class="txt" href="yellow18.html#E3:20"><i><font color="Green">E7</font></i></a>, <a class="ref" href="functor3.html#T35">FUNCTOR3:35</a></i>;<br/>
<a NAME="E12:20"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor3.html#R1">are_naturally_equivalent</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#R1">are_naturally_equivalent</a>  )
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:20"><i><font color="Green">E9</font></i></a>, <a class="txt" href="yellow18.html#E8:20"><i><font color="Green">E10</font></i></a>, <a class="txt" href="yellow18.html#E9:20"><i><font color="Green">E11</font></i></a>, <a class="txt" href="yellow18.html#E10:20"><i><font color="Green">E12</font></i></a>, <a class="ref" href="functor3.html#T36">FUNCTOR3:36</a></i>;<br/>
<b>hence </b><a NAME="E13:20"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>, <a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#R1">are_naturally_equivalent</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>, <a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>3</sub></font> <a href="functor3.html#R1">are_naturally_equivalent</a>  )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:20"><i><font color="Green">E5</font></i></a>, <a class="txt" href="yellow18.html#E4:20"><i><font color="Green">E8</font></i></a>, <a class="ref" href="functor3.html#T33">FUNCTOR3:33</a></i>;<br/>


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