Implication details
Claim: Given a morphism whose category is epi-regular, if it is a monomorphism, then it is a strong monomorphism.
Proof: Any monomorphism is right orthogonal to any regular epimorphism because regular epimorphisms are strong (by combining the dual of this result and the dual of this result). In an epi-regular category, it follows that every monomorphism is right orthogonal to any epimorphism, i.e. a strong monomorphism.
This implication has a dual.