Zero cokernel and epimorphisms in semi-abelian categories

Question

I'm dealing with a question that I can't prove or disprove.

Is it true that in a semi-abelian category (in the sense of Janelidze) a normal monomorphism with null cokernel is an epimorphism?

I know that the fact is trivially true in the category of Grp, since cokernels are quotients, but I don't know if this fact generalizes to the semi-abelian setting.

Answer 1

In a semi-abelian category, a normal monomorphism is the kernel of its cokernel. So if said cokernel is zero, its kernel is an isomorphism, hence also an epimorphism.