Proposition 9.2.6.13. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$, and let $\widehat{\operatorname{\mathcal{C}}}_0 \subseteq \widehat{\operatorname{\mathcal{C}}}$ be the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects. Then $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}_0$ as an idempotent completion of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It follows from Proposition 9.2.3.3 that $h$ induces an equivalence from $\operatorname{\mathcal{C}}$ to a full subcategory of $\widehat{\operatorname{\mathcal{C}}}_{1} \subseteq \widehat{\operatorname{\mathcal{C}}}_0$ which generates $\widehat{\operatorname{\mathcal{C}}}$ under $\lambda $-small $\kappa $-filtered colimits. Applying Lemma 9.2.6.12, we see that every object of $\widehat{\operatorname{\mathcal{C}}}_0$ is a retract of an object of $\widehat{\operatorname{\mathcal{C}}}_{1}$. It will therefore suffice to show that $\widehat{\operatorname{\mathcal{C}}}_0$ is idempotent complete. This follows from Proposition 8.5.4.8, since the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is idempotent complete (Proposition 9.1.9.17) and the full subcategory $\widehat{\operatorname{\mathcal{C}}}_0$ is closed under retracts (Remark 9.2.2.24). $\square$