Proposition 8.5.4.6. Let $\operatorname{\mathcal{C}}$ be an idempotent complete $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that, for every object $X \in \operatorname{\mathcal{C}}_0$ and every object $Y \in \operatorname{\mathcal{C}}$ which is a retract of $X$, there exists an object $Y' \in \operatorname{\mathcal{C}}_0$ which is isomorphic to $Y$. Then $\operatorname{\mathcal{C}}_0$ is idempotent complete.
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Proof. Let $\operatorname{Ret}$ denote the category of Construction 8.5.0.2. Suppose we are given an idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}_0$, carrying the object $\widetilde{X} \in \operatorname{Idem}$ to an object $X = F( \widetilde{X} ) \in \operatorname{\mathcal{C}}_0$. We wish to show that $F$ is a split idempotent in $\operatorname{\mathcal{C}}_0$. Since $\operatorname{\mathcal{C}}$ is idempotent complete, we can extend $F$ to a functor $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$, carrying the object $\widetilde{Y} \in \operatorname{Ret}$ to an object $Y = \overline{F}( \widetilde{Y} )$ which is a retract of $X$. By assumption, we can choose an isomorphism $\alpha _0: Y \rightarrow Y'$, where $Y'$ belongs to $\operatorname{\mathcal{C}}_0$. Using Corollary 4.4.5.3, we can lift $\alpha _0$ to an isomorphism of functors $\alpha : \overline{F} \rightarrow \overline{F}'$ in $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}})$, whose image in $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$ is the identity transformation from $F$ to itself. Then $\overline{F}': \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}_0$ is a splitting of the idempotent $F$. $\square$