Example 9.4.2.11. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and set $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ and $\widehat{\operatorname{\mathcal{D}}} = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$. If $\operatorname{\mathcal{D}}$ is idempotent-complete, then the construction $F \mapsto \operatorname{Ind}_{\kappa }^{\lambda }(F)$ induces an equivalence of $\infty $-categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{c}}( \widehat{\operatorname{\mathcal{C}}}, \widehat{\operatorname{\mathcal{D}}} )$. More precisely, if $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ and $j: \operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$ are functors which exhibit $\widehat{\operatorname{\mathcal{C}}}$ and $\widehat{\operatorname{\mathcal{D}}}$ as $\operatorname{Ind}_{\kappa }^{\lambda }$-completions of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ (respectively), then the functors
are fully faithful and have the same essential image.