Corollary 8.5.5.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Suppose that $\operatorname{\mathcal{D}}$ is a retract of $\operatorname{\mathcal{C}}$ in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$. If $\operatorname{\mathcal{C}}$ is idempotent complete, then $\operatorname{\mathcal{D}}$ is also idempotent complete.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Remark 8.5.3.9, we can identify $\operatorname{\mathcal{D}}$ with the limit of a diagram $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{QC}}$ carrying the unique object of $\operatorname{Idem}$ to the idempotent complete $\infty $-category $\operatorname{\mathcal{C}}$. The desired result is now a special case of Corollary 8.5.5.8. $\square$