Corollary 9.4.2.19 (06HN)

Corollary 9.4.2.19. Let $\kappa \trianglelefteq \lambda \trianglelefteq \mu $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. If $\operatorname{\mathcal{D}}$ is idempotent-complete, then the construction $F \mapsto \operatorname{Ind}_{\lambda }^{\mu }(F)$ induces an equivalence of $\infty $-categories

\[ \operatorname{Fun}^{ (\kappa ,\lambda )-\operatorname{c}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{ (\kappa ,\mu )-\operatorname{c}}( \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}}), \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{D}}) ) \]

Proof. For $\kappa = \lambda $, this is a special case of Example 9.4.2.11 (which does not require the assumption $\lambda \trianglelefteq \mu $). We may therefore assume that $\lambda > \kappa $. In this case, $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are idempotent-complete, and can therefore be identified with the full subcategories of $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ and $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{D}})$ spanned by the $(\lambda ,\mu )$-compact objects (Corollary 9.3.2.8). In this case, the desired result is a reformulation of Proposition 9.4.2.18. $\square$