Suppose that $f: A \to B$ is an epimorphism and $x: X \to B$ is a morphism. Is it true that there exists a morphism $y: X \to A$ such that $f \circ y = x$? Is it necessary that the category is abelian?
I can't see how I might prove this, but also I can't think of an example of an abelian category where it is not true.