Let $g: A \to B$ be a morphism in an abelian category.
Is it true that $g$ is epic iff $im(g)=B$?
Context: Given a short sequence in an abelian category $0 \to A \to B \to C \to 0$ with maps $f: A \to B$ and $g: B \to C$ I have two definition of exactness:
- $Im(0 \to A) = ker(f)$, $Im(f) = ker(g)$ and $Im(g) = Ker(B \to 0)$
- $f$ is monic, $Im(f) = ker(g)$ and $g$ is epic.
I want to prove that they are equivalent, and I would be glad to receive a suggestion on the step above.