Implication Details

Claim: If a category has pullbacks, then it is Cauchy complete.

Proof: A direct proof is possible, but we give a more conceptual one. Suppose that $e : X \to X$ is an idempotent morphism in a category $\C$. It can be regarded as an idempotent morphism $e \to e$ in $\C / X$. If $\C$ has pullbacks, then the slice category $\C / X$ has pullbacks; in fact, the forgetful functor creates all connected limits. The slice category also has a terminal object, and hence is finitely complete. In particular, it has equalizers and is therefore Cauchy complete. Applying the forgetful functor $\C / X \to \C$ to a splitting of $e$ in $\C / X$ yields a splitting of $e$ in $\C$.

This implication has a dual.