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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">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">F3</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">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">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">t'</font> be    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">F3</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="E1:62"/><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="E2:62"/><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">F2</font>,<font color="Maroon">F3</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>thus </b><a NAME="E3:62"/>
( <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> implies <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1"/>
<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>
 ;<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="E3:62_1"/><i><font color="Green">E25</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="E4:62_1"/>
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/>
<b>thus </b><font color="Maroon">F1</font> = 
<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="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><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> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<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="mcart_1.html#K2">`2</a> 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F1'</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<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="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><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> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F2</font>,<font color="Maroon">F3</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>


</div><b>end;</b></div>

<a NAME="E4:62"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:62"/>
( <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2</font> implies <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/>;<br/>


</div>
