 Pseudo-epimorphism

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.