Implication Details

Claim: If a morphism is an epimorphism and is a split monomorphism, then it is an isomorphism.

Proof: Assume that $m : A \to B$ is a split monomorphism, and choose a morphism $e : B \to A$ with $e \circ m = \id_A$. Then $m \circ e \circ m = m = {\id_B} \circ m$. Thus, if $m$ is also an epimorphism, we conclude $m \circ e = \id_B$, showing that $m$ is an isomorphism with inverse $e$.

This implication has a dual.