= Pseudo-epimorphism
{title2=$te=0\Rightarrow t=0$}
A <morphism> $e:A\to B$ in a <pointed category> is pseudo-epic when, for every $t:B\to Z$, the equation $te=0$ forces $t=0$. If a <categorical cokernel> exists, this is equivalent to that cokernel having a <zero object> as codomain. Indeed all zero composites then factor uniquely through zero; conversely the cokernel itself must be zero, and being both zero and epic forces its codomain to have identity zero, hence be a <zero object>. Every <epimorphism> is pseudo-epic. In a <preadditive category>, subtraction shows the converse: $ue=ve$ implies $(u-v)e=0$, hence $u=v$. In a general <pointed category> no subtraction is available.
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