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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="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c5">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/><b>let </b><font color="Maroon" title="c6">G1</font>, <font color="Maroon" title="c7">G2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>;<br/><b>let </b><font color="Maroon" title="c8">s</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font>;<br/><b>let </b><font color="Maroon" title="c9">t</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font>;<br/>

















<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F2</font>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c6">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font>
 ;<br/>

<a NAME="E3:46"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c5">F2</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E1:46"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a></i>;<br/>
<a NAME="E4:46"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">F1</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E2:46"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a></i>;<br/>
<a NAME="E5:46"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c5">F2</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E1:46"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a></i>;<br/>
<a NAME="E6:46"/><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c5">F2</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c5">F2</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E2:46"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a></i>;<br/>
<a NAME="E7:46"/><b>then </b><i><font color="Green" title="E41">A7</font></i>: 
<font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c5">F2</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E5:46"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="nattra_1.html#T25" title="NATTRA_1:th.25">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" title="c10">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>;<br/><a NAME="E1:46_1"/><i><font color="Green" title="E42">A8</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:46"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="isocat_1.html#T30" target="_self" title="ISOCAT_1:th.30">Th30</a></i>;<br/><a NAME="E2:46_1"/><i><font color="Green" title="E43">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:46"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="isocat_1.html#E2:46"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="isocat_1.html#T28" target="_self" title="ISOCAT_1:th.28">Th28</a>, <a class="ref" href="isocat_1.html#T29" target="_self" title="ISOCAT_1:th.29">Th29</a>, <a class="ref" href="cat_1.html#T113" title="CAT_1:th.113">CAT_1:113</a></i>;<br/><a NAME="E3:46_1"/><i><font color="Green" title="E44">A10</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:46"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="isocat_1.html#E2:46"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="isocat_1.html#T28" target="_self" title="ISOCAT_1:th.28">Th28</a>, <a class="ref" href="isocat_1.html#T29" target="_self" title="ISOCAT_1:th.29">Th29</a>, <a class="ref" href="cat_1.html#T113" title="CAT_1:th.113">CAT_1:113</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E5:46"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="isocat_1.html#E6:46"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="isocat_1.html#E2:46_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="nattra_1.html#T27" title="NATTRA_1:th.27">NATTRA_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E2:46"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="isocat_1.html#E1:46_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="isocat_1.html#E3:46_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c4">F1</font></span>)</span></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E3:46"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="isocat_1.html#E4:46"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="nattra_1.html#T27" title="NATTRA_1:th.27">NATTRA_1:27</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:46"/>
<font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K6" title="ISOCAT_1:func.6">(#)</a> <font color="Maroon" title="c8">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c4">F1</font></span>)</span>
 <b>by </b><i><a class="txt" href="isocat_1.html#E7:46"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="isocat_1.html#T31" target="_self" title="ISOCAT_1:th.31">Th31</a></i>;<br/>


</div>
