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<b>let </b><font color="Maroon" title="c1">B</font>, <font color="Maroon" title="c2">C</font>, <font color="Maroon" title="c3">D</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c4">T</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">C</font>;<br/><b>let </b><font color="Maroon" title="c5">S</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">D</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:107"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c4">T</font> is <a href="cat_1.html#V10" title="CAT_1:attr.10">full</a>
 <b>and </b><br/><a NAME="E2:107"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="cat_1.html#V10" title="CAT_1:attr.10">full</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">b'</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D23" target="_self" title="CAT_1:def.23">CAT_1:def 23</a><br/>

<b>assume </b><a NAME="E3:107"/><i><font color="Green" title="E62">A3</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">b'</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> 
 ;<br/>

<b>let </b><font color="Maroon" title="c8">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">b'</font>;<br/>

<a NAME="E4:107"/><i><font color="Green" title="E63">A4</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">b'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T112" target="_self" title="CAT_1:th.112">Th112</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">b'</font><b> such that </b><br/><a NAME="E6:107"/><i><font color="Green" title="E64">A5</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E2:107"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="cat_1.html#E3:107"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="cat_1.html#D23" target="_self" title="CAT_1:def.23">Def23</a></i>;<br/>
<a NAME="E7:107"/><i><font color="Green" title="E65">A6</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">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">b'</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="cat_1.html#E2:107"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="cat_1.html#E3:107"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="cat_1.html#E4:107"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="cat_1.html#D23" target="_self" title="CAT_1:def.23">Def23</a></i>;<br/>
<b>hence </b><a NAME="E8:107"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b'</font> <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="cat_1.html#E1:107"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="cat_1.html#D23" target="_self" title="CAT_1:def.23">Def23</a></i>;<br/>

<b>consider </b><font color="Maroon" title="c10">h</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b'</font><b> such that </b><br/><a NAME="E10:107"/><i><font color="Green" title="E66">A7</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:107"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="cat_1.html#E7:107"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="cat_1.html#D23" target="_self" title="CAT_1:def.23">Def23</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">h</font>
;<br/>

<b>thus </b><a NAME="E11:107"/>
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E6:107"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="cat_1.html#E10:107"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>


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