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Corollary 9.5.5.11. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Then the restriction functor

\[ \operatorname{Fun}^{\lambda -\mathrm{cocont}}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]

is an equivalence of $\infty $-categories.

Proof. By virtue of the universal property of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$, this is a reformulation of Corollary 9.5.5.10. $\square$