Corollary 8.5.8.11. Let $\operatorname{\mathcal{C}}$ be a small filtered $\infty $-category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a functor. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is idempotent complete. Then the colimit $\varinjlim ( \mathscr {F} )$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}$) is also idempotent complete.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Theorem 7.2.7.2, we can choose a directed partially ordered set $(A, \leq )$ and a right cofinal functor $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}$. Using Corollary 7.2.2.3 we can replace $\operatorname{\mathcal{C}}$ by $\operatorname{N}_{\bullet }(A)$ and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is (the nerve of) a directed partially ordered set. Replacing $\mathscr {F}$ by an isomorphic functor if necessary, we can assume that it obtained from an $A$-indexed diagram in the ordinary category $\operatorname{QCat}$ (Corollary 5.6.5.16). In this case, the colimit $\varinjlim ( \mathscr {F} )$ in the $\infty $-category $\operatorname{\mathcal{QC}}$ can be identified with its colimit in the ordinary category $\operatorname{QCat}\subset \operatorname{Set_{\Delta }}$ (Corollary 7.5.9.3), so the desired result follows from Corollary 8.5.8.10. $\square$