Corollary 8.5.8.10. Let $\{ \operatorname{\mathcal{C}}_{i} \} _{i \in \operatorname{\mathcal{I}}}$ be a diagram of simplicial sets indexed by a filtered category $\operatorname{\mathcal{I}}$. Suppose that each $\operatorname{\mathcal{C}}_{i}$ is an idempotent complete $\infty $-category. Then the colimit $\operatorname{\mathcal{C}}= \varinjlim _{i} \operatorname{\mathcal{C}}_{i}$ is idempotent complete.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For each object $i \in \operatorname{\mathcal{I}}$, our assumption that $\operatorname{\mathcal{C}}_{i}$ is idempotent complete guarantees that the restriction functor $R_{i}: \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}_ i ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}_ i )$ is an equivalence of $\infty $-categories (Remark 8.5.4.8). Passing to filtered colimits, we deduce that the induced map $\varinjlim \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}_ i ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}_ i )$ is also an equivalence of $\infty $-categories (Corollary 4.5.7.2). This map fits into a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \varinjlim \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}_ i ) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \ar [d]^{R} \\ \varinjlim \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}_ i ) \ar [r] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}), } \]
where the horizontal maps are equivalences of $\infty $-categories (Corollaries 8.5.8.9 and 8.5.1.32). It follows that the restriction functor $R: \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$ is also an equivalence of $\infty $-categories, so that $\operatorname{\mathcal{C}}$ is idempotent complete (Remark 8.5.4.8). $\square$