Proposition 9.4.6.25. Let $\kappa < \lambda \leq \mu $ be uncountable regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\lambda $-small and locally $\kappa $-small. Then the canonical map $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ exhibits $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ as an idempotent-completion of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Theorem 9.3.4.14, we can identify $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ with the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ spanned by those functors which are $\lambda $-flat and $h_{\bullet }$ with the covariant Yoneda embedding $C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \bullet , C)$. To complete the proof, we must show that every $\lambda $-flat functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is retract of a representable functor. Equivalently, we must show that for each object $C \in \operatorname{\mathcal{C}}$, the Kan complex $\mathscr {F}(C)$ is essentially $\lambda $-small (Remark 9.3.4.12). In fact, we claim that $\mathscr {F}(C)$ is essentially $\kappa $-small. To prove this, we can use Lemma 9.3.4.21 to realize $\mathscr {F}(C)$ as the colimit of a $\mu $-small $\lambda $-filtered diagram
\[ \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet ) } \operatorname{\mathcal{S}}_{< \mu }, \]
which is $\kappa $-small by virtue of Lemma 9.4.6.24 (since $\operatorname{\mathcal{C}}$ is locally $\kappa $-small). $\square$