Corollary 9.2.9.19. Let $\kappa \leq \lambda $ be regular cardinals and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets which is $(\kappa ,\lambda )$-cocomplete. Let $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ denote the $\infty $-category of cocartesian sections of $U$ (Notation 5.3.1.10), and let $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ be an object which carries each vertex $C \in \operatorname{\mathcal{C}}$ to a $(\kappa ,\lambda )$-compact object $F(C) \in \operatorname{\mathcal{E}}_{C}$. If $\operatorname{\mathcal{C}}$ is $\kappa $-small, then $F$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.
Proof. It follows from Corollary 7.1.10.7 that the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ admits $\lambda $-small $\kappa $-filtered colimits, which are preserved by the evaluation functors $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_{C}$ for $C \in \operatorname{\mathcal{C}}$. Using Remark 9.2.9.8 we see that the full subcategory $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is closed under the formation of $\lambda $-small $\kappa $-filtered colimits. Consequently, to show that $F$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, it will suffice to show that it is $(\kappa ,\lambda )$-compact when viewed as an object of the larger $\infty $-category $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. This follows from Proposition 9.2.9.16 (together with Warning 9.2.9.17). $\square$