Converse to: Equivalent conditions for a preabelian category to be abelian

Question

In the following question: Equivalent conditions for a preabelian category to be abelian. How is the converse easily shown? I see why every monomorphism, f, is the kernel of the cokernel of f, but why is every epimorphism, f, the cokernel of the kernel of f?

Answer 1

The kernel of a monomorphism is zero in any pointed category, so the coimage of a mono is the identity. The image is the kernel of the cokernel, so if coimage and image are isomorphic then the map from the domain to the coimage makes the domain a kernel. Dually, we see every dpi is a cokernel.