= Solution
Let $q:B\to Q$ be the <categorical cokernel> of $f$ in an <abelian category>. If $f$ is an <epimorphism>, the equality $qf=0f$ implies $q=0$. Since $q$ is also an <epimorphism>, $1_Qq=0q$ implies $1_Q=0$. An object with zero identity is a <zero object>: every morphism to or from it is zero. Hence the cokernel object is zero.
Conversely, suppose the cokernel object is zero. If $uf=vf$, additivity gives $(u-v)f=0$. The <categorical cokernel> property makes $u-v$ factor through the zero object, so $u-v=0$ and $u=v$. Thus
$$
\boxed{f\text{ is epic}\quad\Longleftrightarrow\quad\operatorname{coker}f=0.}
$$
This is the <zero-cokernel criterion for epimorphisms>. In a <pushout in a category>, the two horizontal morphisms have isomorphic cokernels by part (a). Therefore the lower morphism is epic if and only if the upper morphism is epic. In particular \b[pushouts reflect epimorphisms] in an <abelian category>; the same argument also proves preservation.
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