Corollary 7.4.1.13. Let $\lambda $ be an uncountable cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be the exponential cofinality of $\lambda $. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, where $\operatorname{\mathcal{C}}$ is a $\kappa $-small simplicial set. If the $\infty $-category $\mathscr {F}(C)$ is essentially $\lambda $-small for each $C \in \operatorname{\mathcal{C}}$, then the limit $\varprojlim (\mathscr {F} )$ is also essentially $\lambda $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Proposition 4.7.5.5, we can choose a categorical equivalence $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is a $\lambda $-small $\infty $-category (if $\kappa $ is uncountable, we can even arrange that $\operatorname{\mathcal{D}}$ is $\kappa $-small). Without loss of generality, we may assume that $\mathscr {F}$ is obtained as the restriction of the covariant transport representation of some cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$. Using Corollary 7.4.1.9, we can identify $\varprojlim ( \mathscr {F} )$ with a full subcategory of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. It will therefore suffice to show that the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is essentially $\lambda $-small (Corollary 4.7.5.13). By construction, we have a pullback diagram of simplicial sets
7.46
\begin{equation} \begin{gathered}\label{equation:limit-essential-smallness} \xymatrix@C =50pt@R=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}} ( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [d]^{U \circ } \\ \{ G \} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) } \end{gathered} \end{equation}
where the vertical maps are cocartesian fibrations (Theorem 5.2.1.1), and therefore isofibrations (Proposition 5.1.4.8). It follows that (7.46) is also a categorical pullback square (Corollary 4.5.2.27). Using Corollary 4.7.5.16, we are reduced to proving that the $\infty $-categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ and $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ are essentially $\lambda $-small, which follows from Remark 4.7.5.10. $\square$