Corollary 2.39.
Any morphism which is both a monomorphism and an effective epimorphism is an isomorphism. In particular, every monomorphism with a section is an isomorphism.
Proof
For the first claim, recall that if a category \(C\) is equipped with a factorization system \((L,R)\), then any morphism \(f\colon X \to Y\) in both \(L\) and \(R\) is an isomorphism (see Lemma A.4). The second claim follows immediately from the first combined with Lemma 2.38.