Implication Details

Claim: If a category is regular-subobject-trivial, then it has reflexive coequalizers.

Proof: Let $f,g : X \rightrightarrows Y$ be a reflexive pair, i.e. there is a morphism $r : Y \to X$ with $f r = g r = \id_Y$. Since $r$ is a split monomorphism, it is a regular monomorphism. By assumption, $r$ must be an isomorphism. Thus, $f = g = r^{-1}$, and $\id_Y$ is a coequalizer of $f,g$.

This implication has a dual.