Implication Details

Claim: If a morphism is a split monomorphism, then it is a regular monomorphism.

Proof: Let $m : A \to B$ be a split monomorphism, and choose a morphism $e : B \to A$ with $e \circ m = \id_A$. Then it is easy to check that $m$ is an equalizer of $\id_B$ and the idempotent morphism $m \circ e : B \to B$.