Implication details

Proof: Let $q_1,q_2 : R \rightrightarrows X$ be a congruence. In particular, there is a morphism $r : X \to R$ with $q_1 r = q_2 r = \id_X$. Since $r$ is a split monomorphism, $r$ must be an isomorphism. Thus, $q_1 = q_2 = r^{-1}$, and $q_1,q_2$ is a kernel pair of $\id_X$.