Implication Details

Claim: If a category is regular-subobject-trivial, then it has effective congruences.

Proof: Let $f,g : E \rightrightarrows X$ be a congruence. In particular, there is a morphism $r : X \to E$ with $f r = g r = \id_X$. Since $r$ is a split monomorphism, it is a regular monomorphism. By assumption, $r$ must be an isomorphism. Thus, $f = g = r^{-1}$, and $f,g$ is a kernel pair of $\id_X$.

This implication has a dual.