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Corollary 8.5.4.17. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which admits sequential limits (or colimits). Then $\operatorname{\mathcal{C}}$ is idempotent complete.

Proof. It follows from Proposition 8.5.4.16 (and Corollary 7.2.2.12) that the $\infty$-category $\operatorname{\mathcal{C}}$ admits limits (or colimits) indexed by the $\infty$-category $\operatorname{N}_{\bullet }( \operatorname{Idem})$, and is therefore idempotent complete by virtue of Proposition 8.5.4.7. $\square$