surjective map as epimorphism

Let be a concrete category and a morphism, such that is surjective (cf. underlying-set functor) Then is a epimorphism

Suppose there exists an object and morphisms such that

commutes

Then applying the set functor results in the commuting diagram

where by assumption of surjectivity, for , there exists an such that

Then by commutativity, it follows, that

and since was arbitrary

Hence

commutes and by faithful functor and preimage as diagram also

Thus

and we conclude, that , showing that is mono