Must an epimorphism in abelian category have cokernel $0$?

Question

Suppose $\mathcal{A}$ is an abelian category, that is an additive category with

1) a zero object,

2) all binary products and binary coproducts,

3) all kernels and cokernels.

4) monomorphisms are kernel of a morphism and epimorphisms are cokernel of a morphism.

Must an epimorphism in abelian category have cokernel $0$? Suppose we have $A\to B\to \text{Coker}(f)$, is the latter map epic? Must the latter map has cokernel $0$?

Answer 1

In any category with zero object is is verified easily that an epimorphism $f : A \to B$ has cokernel $0 : B \to 0$. In fact, $0f=0$, and if $g : B \to C$ is a morphism such that $gf=0$, then $g=0$ (since $f$ is epi) and therefore $g$ factors uniquely through $0$.

Answer 2

Hint: For the second question, in any category, every coequalizer is an epimorphism. And cokernel is the coequalizer of $f$ and the zero morphism.