Implication details

Proof: Let $f,g : X \rightrightarrows Y$ be a coreflexive pair, i.e. there is a morphism $r : Y \to X$ with $rf = rg = \id_X$. Then $r$ is a split and hence a regular epimorphism. Thus, $r$ is an isomorphism. But then $f=g$, and $\id_X$ is an equalizer of $f,g$. This equalizer is obviously preserved.