Corollary 8.5.4.25. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. If $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ are idempotent complete, then the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$ is also idempotent complete.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It follows from Proposition 8.5.1.7 that $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$ is closed under retracts in the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. The desired result now follows from Propositions 8.5.4.23 and 8.5.4.8. $\square$