Remark 8.5.4.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is idempotent complete if and only if the restriction functor
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}) \]
is an equivalence of $\infty $-categories. This is an immediate consequence of Corollary 8.5.3.10.