This is an exercise in Ravil Vakil's Algebraic Geometry notes labeled hard
Let $f: A\rightarrow B$ be a morphism in an abelian category. Then $f$ factors through a morphism $f':A\rightarrow \operatorname{im}(f)$ (recall that $\operatorname{im}(f)$ is the (domain of the) kernel of the cokernel of $f$).
Why is $f'$ an epimorphism? That is why does $g\circ f'=0$ imply $g=0$ for any $g: \operatorname{im}(f)\rightarrow C$ ?