Kerodon

Corollary 9.2.8.11 (Compactness in Coslice $\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 $\kappa $-small diagram which carries each vertex of $K$ to a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$. If $X$ is an object of the coslice $\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 $\operatorname{\mathcal{C}}_{F/}$.