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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E53">Th58</font></span>: <a NAME="T58"><span class="comment"><font color="firebrick">:: CAT_8:58</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b3">F1</font>, <font color="Olive" title="b4">F2</font>, <font color="Olive" title="b5">T</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">C1</font>,<font color="Olive" title="b2">C2</font>  st <font color="Olive" title="b3">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b4">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/>( <font color="Olive" title="b5">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Olive" title="b3">F1</font>,<font color="Olive" title="b4">F2</font> iff  for <font color="Olive" title="b6">f</font>, <font color="Olive" title="b7">f1</font>, <font color="Olive" title="b8">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Olive" title="b1">C1</font>  st <font color="Olive" title="b7">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b8">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b6">f</font> &amp; <font color="Olive" title="b6">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b8">f2</font> holds <br/>( <font color="Olive" title="b5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f</font> &amp; <font color="Olive" title="b4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b8">f2</font> &amp; <font color="Olive" title="b5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b7">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f</font></span>)</span> &amp; <font color="Olive" title="b5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b8">f2</font></span>)</span> ) )</div></div>
