Pullback stability of epimorphisms in an abelian category

In an abelian category, the pullback in a category of an epimorphism $p:B\to C$ along $k:C^{\prime }\to C$ is the kernel of the epimorphism $[p,-k]:B\oplus C^{\prime }\to C$. If $v:C^{\prime }\to D$ annihilates its projection, $[0,v]$ annihilates that kernel and therefore factors through $[p,-k]$. Restriction to $B$ forces the factor to vanish, giving $v=0$ and proving the projection epic. This uses the definition's normality of epimorphisms.

In an abelian category, suppose a commutative square $gf=kh$ is a pullback in a category and $g$ is an epimorphism. Then $(f,h):A\to B\oplus C$ is the categorical kernel of the epimorphism $[g,-k]$. Since an epimorphism is the cokernel of its kernel, any compatible pair $u:B\to X$, $v:C\to X$ induces a unique map $D\to X$ by factoring $[u,-v]$. Hence the square is also a pushout in a category.