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<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c4">D</font> be   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">F2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C2</font>,<font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b3">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b4">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b4">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">D</font>,<font color="Olive" title="b3">P1</font>,<font color="Olive" title="b4">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> &amp; <font color="Olive" title="b1">F1</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  holds <br/><font color="Olive" title="b4">P2</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">F1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c1">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C2</font>,<font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b3">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c5">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">D</font>,<font color="Olive" title="b2">P1</font>,<font color="Olive" title="b3">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c5">F1</font>,<font color="Olive" title="b1">F2</font> &amp; <font color="Maroon" title="c5">F1</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  holds <br/><font color="Olive" title="b3">P2</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">F2</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C2</font>,<font color="Maroon" title="c1">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b2">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c5">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">D</font>,<font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font> &amp; <font color="Maroon" title="c5">F1</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  holds <br/><font color="Olive" title="b2">P2</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c7">P1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">C1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Maroon" title="c5">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c7">P1</font>,<font color="Olive" title="b1">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font> &amp; <font color="Maroon" title="c5">F1</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  holds <br/><font color="Olive" title="b1">P2</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c8">P2</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">C2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c8">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c8">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font> &amp; <font color="Maroon" title="c5">F1</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  implies <font color="Maroon" title="c8">P2</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  )</span><br/>









<span class="kw">assume </span><a NAME="E1:81"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
( <font color="Maroon" title="c5">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c8">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c8">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font> or  not <font color="Maroon" title="c5">F1</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  or <font color="Maroon" title="c8">P2</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:81"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c8">P2</font> <a href="cat_7.html#R3" title="CAT_7:pred.3">is_pullback_of</a> <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">F1</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  or <font color="Maroon" title="c8">P2</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  )</span><br/>

<a NAME="E3:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A3</font></span>: 
( <font color="Maroon" title="c5">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c7">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font> &amp; (  for <font color="Olive" title="b1">D1</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c2">C1</font><br/>  for <font color="Olive" title="b3">G2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b2">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">G2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c5">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b3">G2</font> holds <br/> ex <font color="Olive" title="b4">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c4">D</font> st <br/>( <font color="Olive" title="b4">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c8">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> &amp; (  for <font color="Olive" title="b5">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c4">D</font>  st <font color="Olive" title="b5">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G1</font> &amp; <font color="Maroon" title="c8">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b5">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">G2</font> holds <br/><font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">H1</font> ) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_7.html#D20" target="_self" onmouseover="rs('cat_7/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:81"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c5">F1</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">P2</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">C1</font><span class="kw"> such that </span><br/><a NAME="E6:81"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
( <font color="Maroon" title="c9">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c9">G1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c5">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c5">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">G1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E47">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">G11</font> = <font color="Maroon" title="c9">G1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">G22</font> =  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c3">C2</font>;<br/>
<a NAME="E7:81"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<font color="Maroon" title="c9">G1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>;<br/>
<span class="lab"><font color="Green" title="E50">A7</font></span>: <font color="Maroon" title="c5">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">G1</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_7.html#T10" target="_self" onmouseover="rs('cat_7/T10')" onmouseout="rh()">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c6">F2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_7.html#T11" target="_self" onmouseover="rs('cat_7/T11')" onmouseout="rh()">Th11</a></span>
<br/>.= 
<font color="Maroon" title="c6">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c3">C2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_7.html#T11" target="_self" onmouseover="rs('cat_7/T11')" onmouseout="rh()">Th11</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">Q2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C2</font>,<font color="Maroon" title="c4">D</font><span class="kw"> such that </span><br/><a NAME="E10:81"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
( <font color="Maroon" title="c12">Q2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">Q2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">G1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F2</font> &amp; <font color="Maroon" title="c8">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">Q2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c3">C2</font> &amp; (  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C2</font>,<font color="Maroon" title="c4">D</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">G1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F2</font> &amp; <font color="Maroon" title="c8">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c3">C2</font> holds <br/><font color="Maroon" title="c12">Q2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:81"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_7.html#D20" target="_self" onmouseover="rs('cat_7/D20')" onmouseout="rh()">Def20</a>, <a class="txt" href="#E8:81"><span class="lab"><font color="Green" title="E50">A7</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">G33</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">Q2</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font>;<br/>
<a NAME="E11:81"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c6">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>;<br/>
<a NAME="E12:81"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">Q2</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:81"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E7:81"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>;<br/>
<font color="Maroon" title="c5">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">Q2</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font></span>)</span> = 
<font color="Maroon" title="c5">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E10:81"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_7.html#T10" target="_self" onmouseover="rs('cat_7/T10')" onmouseout="rh()">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">G1</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:81"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="cat_7.html#T10" target="_self" onmouseover="rs('cat_7/T10')" onmouseout="rh()">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c6">F2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:81"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="cat_7.html#T11" target="_self" onmouseover="rs('cat_7/T11')" onmouseout="rh()">Th11</a>, <a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E48">A5</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c4">D</font><span class="kw"> such that </span><br/><a NAME="E15:81"/>
( <font color="Maroon" title="c14">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c14">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">Q2</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font> &amp; <font color="Maroon" title="c8">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c14">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">P2</font> )
 <span class="kw">and </span><br/><a NAME="E16:81"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
 for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c4">D</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">Q2</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font> &amp; <font color="Maroon" title="c8">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">P2</font> holds <br/><font color="Maroon" title="c14">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E12:81"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="ref" href="cat_7.html#D20" target="_self" onmouseover="rs('cat_7/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<span class="lab"><font color="Green" title="E55">A12</font></span>: <font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c4">D</font></span>)</span> = 
<font color="Maroon" title="c7">P1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_7.html#T11" target="_self" onmouseover="rs('cat_7/T11')" onmouseout="rh()">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">G1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c5">F1</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c7">P1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="cat_7.html#T11" target="_self" onmouseover="rs('cat_7/T11')" onmouseout="rh()">Th11</a></span>
<br/>.= 
<font color="Maroon" title="c9">G1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c7">P1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="cat_7.html#T10" target="_self" onmouseover="rs('cat_7/T10')" onmouseout="rh()">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">Q2</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:81"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E3:81"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="cat_7.html#T10" target="_self" onmouseover="rs('cat_7/T10')" onmouseout="rh()">Th10</a></span>
;<br/>
<a NAME="E18:81"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
<font color="Maroon" title="c8">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cat_7.html#T11" target="_self" onmouseover="rs('cat_7/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<a NAME="E19:81"/><span class="lab"><font color="Green" title="E57">A14</font></span>: 
<font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">Q2</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E10:81"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="cat_7.html#T10" target="_self" onmouseover="rs('cat_7/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<font color="Maroon" title="c8">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c12">Q2</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E10:81"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="cat_7.html#T10" target="_self" onmouseover="rs('cat_7/T10')" onmouseout="rh()">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c8">P2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E10:81"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="cat_7.html#T11" target="_self" onmouseover="rs('cat_7/T11')" onmouseout="rh()">Th11</a></span>
;<br/>
<a NAME="E21:81"/><span class="kw">then </span>
<font color="Maroon" title="c14">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:81"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E19:81"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E10:81"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:81"/>
<font color="Maroon" title="c8">P2</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:81"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E18:81"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E16:81"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E17:81"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E44">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
