Corollary 9.2.8.2 (Compactness in Slice $\infty $-Categories). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be a $(\kappa ,\lambda )$-cocomplete $\infty $-category, and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. If $X$ is an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/F}$ whose image in $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact, then $X$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{C}}_{/F}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It follows from Corollary 7.1.4.27 that the $\infty $-category $\operatorname{\mathcal{C}}_{/F}$ admits $\lambda $-small $\kappa $-filtered colimits which are preserved by the right fibration $U: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}$, so the desired result is a special case of Proposition 9.2.8.1. $\square$