Implication Details

Claim: If a category is multi-complete, then it is Cauchy complete.

Proof: More precisely, if a category has multi-equalizers, then it is Cauchy complete. Namely, if $e : X \to X$ is an idempotent morphism, a cone over $e,\id_X$ is a morphism $f : Y \to X$ with $f = e \circ f$. In particular, $e : X \to X$ is such a cone, and $f$ can be regarded as a morphism of cones $f \to e$. This shows that the category of cones is connected. A multi-terminal object in the category of cones is therefore a terminal object, i.e. an equalizer of $e,\id_X$, and hence a splitting of $e$ (cf. this result).