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 $0$?