Lemma 9.5.5.16 (06P1)

Lemma 9.5.5.16. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which satisfies the following conditions:

$(0)$: The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ admits $\lambda $-small colimits.
$(1)$: The functor $h$ is fully faithful and preserves $\kappa $-small colimits.
$(2)$: For each object $C \in \operatorname{\mathcal{C}}$, the image $h(C)$ is a $(\kappa ,\lambda )$-compact object of $\widehat{\operatorname{\mathcal{C}}}$.

Then the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of $h$ is a fully faithful functor $H: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \widehat{\operatorname{\mathcal{C}}}$, whose essential image is the smallest full subcategory which contains the essential image of $h$ and is closed under $\lambda $-small colimits.

Proof. Proposition 9.3.2.1 guarantees that $H$ is fully faithful, and therefore restricts to an equivalence of $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{C}})$ with a replete full subcategory $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$. Since $h$ is $\kappa $-cocontinuous, Corollary 9.5.5.10 guarantees that $H$ is $\lambda $-cocontinuous: that is, the full subcategory $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ is closed under $\lambda $-small colimits. We conclude by observing that every object of $\widehat{\operatorname{\mathcal{C}}}'$ can be realized as the colimit of a $\lambda $-small diagram in $\operatorname{\mathcal{C}}$ (which we can even take to be $\kappa $-filtered; see Corollary 9.3.4.17). $\square$