Proposition 9.2.9.1. Let $\kappa \leq \lambda $ and $\mu $ be regular cardinals, where $\mu $ is uncountable. Let $\operatorname{\mathcal{D}}$ be the limit of a $\kappa $-small diagram
\[ \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{(\kappa ,\lambda )-\mathrm{ccomp}}_{< \mu } \quad \quad (C \in \operatorname{\mathcal{C}}) \mapsto \mathscr {F}(C). \]
Let $D \in \operatorname{\mathcal{D}}$ be an object having the property that, for each $C \in \operatorname{\mathcal{C}}$, the image of $D$ in $\mathscr {F}(C)$ is $(\kappa ,\lambda )$-compact. Then $D$ is $(\kappa ,\lambda )$-compact.
Proof of Proposition 9.2.9.1.
Let $\kappa \leq \lambda $ and $\mu $ be regular cardinals, where $\mu $ is uncountable, and let $\operatorname{\mathcal{D}}$ be the limit of a $\kappa $-small diagram
\[ \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{(\kappa ,\lambda )-\mathrm{ccomp}}_{< \mu } \quad \quad (C \in \operatorname{\mathcal{C}}) \mapsto \mathscr {F}(C) \]
Then $\mathscr {F}$ is the covariant transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ which is essentially $\mu $-small and $(\kappa ,\lambda )$-cocomplete (Example 7.6.6.33). Using Proposition 7.4.4.1, we can identify $\operatorname{\mathcal{D}}$ with the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian sections of $U$. Under this identification, each of the projection maps $\operatorname{\mathcal{D}}\rightarrow \mathscr {F}(C)$ corresponds to the functor
\[ \operatorname{ev}_{C}: \operatorname{Fun}_{ /\operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \{ C \} , \operatorname{\mathcal{E}}) = \operatorname{\mathcal{E}}_{C} \]
given by evaluation at $C$. Applying Corollary 9.2.9.19, we see that if an object $D \in \operatorname{\mathcal{D}}$ has $(\kappa ,\lambda )$-compact image in each $\mathscr {F}(C)$, then $D$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{D}}$.
$\square$