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Proposition 9.2.4.22. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\lambda $-small. Let $\widehat{\operatorname{\mathcal{C}}} \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ be the full subcategory spanned by those functors $\mathscr {F}$ which satisfy the following condition:

$(\ast )$

There exists a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ such that $\mathscr {F}|_{ \operatorname{\mathcal{C}}^{\operatorname{op}}_0 }$ is $\kappa $-flat, $\mathscr {F}$ is left Kan extended from $\operatorname{\mathcal{C}}^{\operatorname{op}}_{0}$, and $\operatorname{\mathcal{C}}_0$ is essentially $\lambda $-small.

Then the covariant Yoneda embedding

\[ h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}} \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) \]

exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$.

Proof. Using Proposition 8.4.5.8, we can identify $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ with the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ which contains all representable functors and is closed under $\lambda $-small $\kappa $-filtered colimits. For every full subcategory $\operatorname{\mathcal{C}}_0$ of $\operatorname{\mathcal{C}}$, the inclusion functor $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ induces a functor

\[ \operatorname{Ind}_{\kappa }^{\lambda }(\iota ): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}_0) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \]

which is given (under the preceding identification) by left Kan extension along $\iota $ (Remark 9.2.1.21). Using Theorem 9.2.4.14, we see that a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ is contained in $\widehat{\operatorname{\mathcal{C}}}$ if and only if it belongs to the essential image of $\operatorname{Ind}_{\kappa }^{\lambda }(\iota )$ for some full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ which is essentially $\lambda $-small. This follows from Proposition 9.2.1.23, since the collection of such subcategories comprise a $\lambda $-filtered diagram having colimit $\operatorname{\mathcal{C}}$. $\square$