Implication Details

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

Proof: Assume that $e : X \to X$ is an idempotent morphism. Consider the sequence $$X \xrightarrow{e} X \xrightarrow{e} X \to \cdots.$$ A cocone under this sequence is a family of morphisms $f_n : X \to Y$ satisfying $$f_n = f_{n+1} e.$$ Then $$f_n = f_{n+1} e = f_{n+2} e^2 = f_{n+2} e = f_{n+1}$$ shows that all the morphisms are equal. Thus, a cocone is the same as a morphism $f_0 : X \to Y$ with $f_0 = f_0 e$, meaning it coequalizes $\id_X,e : X \rightrightarrows X$. Hence, if a colimit exists, $e$ splits.