Corollary 9.2.9.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa \trianglelefteq \lambda $ be small regular cardinals, where $\lambda $ is uncountable. Then an object $X \in \operatorname{Ind}_{\kappa }(\operatorname{\mathcal{C}})$ is $\lambda $-compact if and only if it belongs to the full subcategory $\operatorname{Ind}^{\lambda }_{\kappa }(\operatorname{\mathcal{C}})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Apply Corollary 9.2.9.7 in the special case where $\mu = \operatorname{\textnormal{\cjRL {t}}}$ is strongly inaccessible. $\square$