Implication details
Claim: Given a morphism whose category is mono-regular, if it is a monomorphism, then it is a regular monomorphism.
Proof: This is the definition of a mono-regular category.
This implication has a dual.
Claim: Given a morphism whose category is mono-regular, if it is a monomorphism, then it is a regular monomorphism.
This implication has a dual.