Proposition 9.1.10.4. Let $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}$ be a filtered diagram of $\infty $-categories. Suppose that each of the $\infty $-categories $\mathscr {F}(J)$ is idempotent complete. Then the colimit $\varinjlim (\mathscr {F})$ is also idempotent complete.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. As in the proof of Proposition 9.1.10.1, we can assume that $\operatorname{\mathcal{J}}= \operatorname{N}_{\bullet }(A)$ is the nerve of a directed partially ordered set $(A, \leq )$ and that $\mathscr {F}$ is obtained from a strictly commutative diagram $\mathscr {F}_0: (A, \leq ) \rightarrow \operatorname{QCat}\subset \operatorname{Set_{\Delta }}$. Using Corollary 9.1.6.3 , we are reduced to proving that the colimit $\varinjlim (\mathscr {F}_0)$ (formed in the category of simplicial sets) is idempotent complete, which follows from Corollary 8.5.9.10. $\square$