Corollary 9.3.4.15. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\lambda $-small, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$. Then, for every object $X \in \widehat{\operatorname{\mathcal{C}}}$, the $\infty $-category $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\times _{ \widehat{\operatorname{\mathcal{C}}} } \widehat{\operatorname{\mathcal{C}}}_{/X}$ is $\kappa $-filtered and essentially $\lambda $-small. Moreover, the tautological map $\operatorname{\mathcal{E}}^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{C}}}_{/X}^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{C}}}$ is a colimit diagram. In particular, the object $X$ is a colimit of the essentially $\lambda $-small $\kappa $-filtered diagram $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}\xrightarrow {h} \widehat{\operatorname{\mathcal{C}}}$.
Proof. By virtue of Theorem 9.3.4.14, we may assume without loss of generality that $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{<\lambda })$ and that $h$ is the covariant Yoneda embedding. In this case, we can identify $X$ with a $\kappa $-flat functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$, so that the projection map $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is an essentially $\lambda $-small right fibration with contravariant transport representation $\mathscr {F}$ (Corollary 8.4.2.7). Since $\mathscr {F}$ is $\kappa $-flat, the $\infty $-category $\operatorname{\mathcal{E}}$ is $\kappa $-filtered. Since $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, Corollary 4.9.8.12 guarantees that $\operatorname{\mathcal{E}}$ is also essentially $\lambda $-small. The last assertion follows from the density of the covariant Yoneda embedding (Variant 8.4.2.4). $\square$