Epimorphisms in an abelian category

Question

Let $C$ be an abelian category.

Let $f:X \to Y$ and $g: Y \to Z$ are morphisms of $C$ and suppose that $g\circ f$ is epimorphism.

Question Is $g$ epi as well?

If so, I want to know the proof.

Answer 1

Let $r,p:Z\to W$ be morphisms such that $r\circ g=p\circ g$. Then $r\circ g\circ f=p\circ g\circ f$, which implies that $r=p$ since $g\circ f$ is epi. Thus, $g$ is also epi.

Note that I didn't use the abelianess of the category, so it holds in arbitrary categories.