Implication Details

Claim: If a category has coequalizers of kernel pairs, then it is Cauchy complete.

Proof: Let $e : X \to X$ be an idempotent morphism. By assumption, the kernel pair $p_1,p_2 : E \rightrightarrows X$ of $e$ exists, i.e. the universal pair satisfying $e \circ p_1 = e \circ p_2$, and it has a coequalizer $q : X \to Q$. The morphism $e : X \to X$ coequalizes $p_1,p_2$, so there is a morphism $i : Q \to X$ satisfying $e = i \circ q$. It remains to prove that $q \circ i = \id_Q$. Since $q$ is an epimorphism, it suffices to prove that $q \circ i \circ q = q$, i.e. that $q \circ e = q$.
Because $e \circ e = e \circ \id_X$, the universal property of the kernel pair yields a (unique) morphism $f : X \to E$ with $p_1 \circ f = e$ and $p_2 \circ f = \id_X$. Since $q \circ p_1 = q \circ p_2$, we obtain $$q \circ e = q \circ p_1 \circ f = q \circ p_2 \circ f = q,$$ finishing the proof.

This implication has a dual.