Implication details

Proof: Let $m : A \to B$ be an extremal monomorphism in a coregular category. By coregularity, we may factor it as $m = i \circ e$, where $i : C \to B$ is a regular monomorphism and $e : A \to C$ is an epimorphism. Since $m$ is an extremal monomorphism, $e$ is an isomorphism. Therefore, $m \cong i$ is a regular monomorphism.
The proof shows that the assumption of coregularity can be relaxed to the existence of (Epi, RegMono)-factorizations.