Implication Details
Claim: Given a functor whose codomain is regular-subobject-trivial, if it is dominant, then it is essentially surjective.
Proof: This is trivial since every split monomorphism in the codomain is an isomorphism.
Claim: Given a functor whose codomain is regular-subobject-trivial, if it is dominant, then it is essentially surjective.
Proof: This is trivial since every split monomorphism in the codomain is an isomorphism.