Corollary 9.3.7.13. Let $\kappa \trianglelefteq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\kappa $-filtered $\infty $-categories. Then $F$ is right cofinal if and only if the induced functor $\operatorname{Ind}_{\kappa }^{\lambda }(F): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ is right cofinal.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Assume that $F$ is right cofinal; we will show that $\widehat{F} = \operatorname{Ind}_{\kappa }^{\lambda }(F)$ is right cofinal (the converse is a special case of Lemma 9.3.7.10, and requires weaker assumptions). Choose a regular cardinal $\mu $ satisfying $\lambda \trianglelefteq \mu $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\mu $-small. By virtue of Lemma 9.3.7.10, it will suffice to show that $\operatorname{Ind}_{\lambda }^{\mu }( \widehat{F} )$ is right cofinal. This follows from Proposition 9.3.7.11, because we can identify $\operatorname{Ind}_{\lambda }^{\mu }( \widehat{F} )$ with the functor $\operatorname{Ind}_{\kappa }^{\mu }(F): \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{D}})$ (see Theorem 9.3.6.4 and Proposition 9.2.2.29). $\square$