It is an 'if and only if', so we would need an isomorphism (epimorphism gives the 'if', monomorphism gives the 'only if').
Why bother with the monomorphism bit? If is an isomorphism then it is an epimorphism and so the above proof works, and since 1is an epimorphism too the conclusion follows.
Why bother with the monomorphism bit? If is an isomorphism then it is an epimorphism and so the above proof works, and since 1is an epimorphism too the conclusion follows.
One way is as easy as the other. I explained it in that way as I had maybe thought you had meant monomorphism as opposed to epimorphism, so I had thought about both cases.