Corollary 9.2.8.17. Let $\kappa $ and $\lambda $ be uncountable regular cardinals satisfying $\kappa \trianglelefteq \lambda $. Then the inclusion map $\operatorname{\mathcal{QC}}_{< \kappa } \hookrightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ exhibits $\operatorname{\mathcal{QC}}_{< \lambda }$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{QC}}_{< \kappa }$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Note that $\operatorname{\mathcal{QC}}_{< \lambda }$ admits $\lambda $-small colimits (Corollary 7.4.5.22), and that objects of $\operatorname{\mathcal{QC}}_{< \kappa }$ are $(\kappa ,\lambda )$-compact when viewed as objects of $\operatorname{\mathcal{QC}}_{< \lambda }$ (Proposition 9.2.8.16). By virtue of Proposition 9.2.3.3, it will suffice to show that every $\lambda $-small $\infty $-category $\operatorname{\mathcal{C}}$ can be realized as the colimit of a $\lambda $-small, $\kappa $-filtered colimit $\{ \operatorname{\mathcal{C}}_{\alpha } \} $, where each $\operatorname{\mathcal{C}}_{\alpha }$ is $\kappa $-small. This follows from Variant 9.1.7.21 and Corollary 9.1.6.3. $\square$