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<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/>

<b>thus </b><a NAME="E1:27"/>
 <a href="altcat_2.html#K5" target="_self">the_comps_of</a> <font color="Maroon">C</font> is <a href="altcat_1.html#V5">with_left_units</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9">ALTCAT_1:def 9</a><br/>

<b>take </b>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">i</font>
;<br/>

<b>thus </b><a NAME="E1:27_1"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">i</font>
 <b>by </b><i/>;<br/>

<a NAME="E2:27_1"/><i><font color="Green">E16</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">i</font>,<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T56">CAT_1:56</a></i>;<br/>
<b>let </b><font color="Maroon">j</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E3:27_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>
 ;<br/>

<a NAME="E4:27_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">f</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">j</font>,<font color="Maroon">i</font> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K5" target="_self">the_comps_of</a> <font color="Maroon">C</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>,<font color="Maroon">i</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">i</font></span>)</span> <a href="cat_1.html#K9">*</a> <font color="Maroon">m</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T57">CAT_1:57</a></i>
;<br/>


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

<b>let </b><font color="Maroon">j</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8">ALTCAT_1:def 8</a><br/>

<b>take </b>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">j</font>
;<br/>

<b>thus </b><a NAME="E2:27"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">j</font>
 <b>by </b><i/>;<br/>

<a NAME="E3:27"/><i><font color="Green">E20</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">j</font>,<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T56">CAT_1:56</a></i>;<br/>
<b>let </b><font color="Maroon">i</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E4:27"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>
 ;<br/>

<a NAME="E5:27"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">f</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">j</font>,<font color="Maroon">i</font> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K5" target="_self">the_comps_of</a> <font color="Maroon">C</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">j</font>,<font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">j</font></span>)</span> = 
<font color="Maroon">m</font> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T58">CAT_1:58</a></i>
;<br/>


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