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

<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">F1</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">F2</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">G1</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">G2</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font>;<br/>

















<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">G1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font>
 ;<br/>

<a NAME="E3:46"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:46"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:46"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:46"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:46"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#T25">NATTRA_1:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><a NAME="E1:46_1"/><i><font color="Green">E57</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E2:46_1"/><i><font color="Green">E59</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="isocat_1.html#E3:46"><i><font color="Green">E22</font></i></a>, , <a class="ref" href="cat_1.html#T113">CAT_1:113</a></i>;<br/><a NAME="E3:46_1"/><i><font color="Green">E60</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="nattra_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="isocat_1.html#E3:46"><i><font color="Green">E22</font></i></a>, , <a class="ref" href="cat_1.html#T113">CAT_1:113</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="nattra_1.html#T27">NATTRA_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="isocat_1.html#T8" target="_self">Th8</a>, <a class="ref" href="nattra_1.html#T27">NATTRA_1:27</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E9:46"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="isocat_1.html#T9" target="_self">Th9</a>, <a class="txt" href="isocat_1.html#E5:46"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:46"/>
<font color="Maroon">t</font> <a href="isocat_1.html#K6" target="_self">(#)</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F1</font></span>)</span>
 ;<br/>


</div>
