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<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">F</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">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">t</font> be    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F</font>,<font color="Maroon">F1</font>;<br/><b>let </b><font color="Maroon">t1</font> be    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" target="_self">Functors</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" target="_self">Functors</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E3:63"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" target="_self">Functors</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">F'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F'</font>,<font color="Maroon">F1'</font>, <font color="Maroon">t1'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E5:63"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1'</font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E6:63"/><i><font color="Green">E27</font></i>: 
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" target="_self">Functors</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1'</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E7:63"/><i><font color="Green">E33</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">t</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t'</font> &amp; <font color="Maroon">t1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t1'</font> )
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:63"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F'</font> &amp; <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F1'</font> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2'</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E9:63"/>
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , , <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>;<br/>


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