Proposition 9.3.3.7. Let $\kappa \leq \lambda < \mu $ be regular cardinals. If $\mu $ has exponential cofinality $\geq \lambda $, then there exists an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion functor $T: \operatorname{\mathcal{QC}}_{< \mu } \rightarrow \operatorname{\mathcal{QC}}_{< \mu }$. Moreover, $T$ is uniquely determined up to isomorphism.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue Corollary 8.4.7.9 (and Remark 9.3.3.6), it will suffice to observe that if $\operatorname{\mathcal{C}}$ an essentially $\mu $-small $\infty $-category, then $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is also essentially $\mu $-small, which follows from Variant 9.3.3.3. $\square$