$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 9.2.8.7 (Compactness in Sequential Limits). Let $\kappa \leq \lambda $ be uncountable regular cardinals and suppose we are given a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{\mathcal{C}}_{3} \xrightarrow { U_2} \operatorname{\mathcal{C}}_2 \xrightarrow { U_1 } \operatorname{\mathcal{C}}_1 \xrightarrow {U_0} \operatorname{\mathcal{C}}_0. \]
Assume that:
Each of the $\infty $-categories $\operatorname{\mathcal{C}}_{n}$ is $(\kappa ,\lambda )$-cocomplete.
Each of the functors $U_{n}: \operatorname{\mathcal{C}}_{n+1} \rightarrow \operatorname{\mathcal{C}}_ n$ is $(\kappa ,\lambda )$-finitary.
Each of the functors $U_ n$ is an isofibration, so that the inverse limit $\varprojlim (\operatorname{\mathcal{C}}_{n})$.
Let $C$ be an object of the inverse limit $\varprojlim \operatorname{\mathcal{C}}_{n}$, given by a sequence of objects $\{ C_ n \in \operatorname{\mathcal{C}}_ n \} $ satisfying $F_{n}(C_{n+1} ) = C_{n}$. If each $C_{n}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{n}$, then $C$ is $(\kappa ,\lambda )$-compact object of $\varprojlim (\operatorname{\mathcal{C}}_{n})$.
Proof.
Set $\operatorname{\mathcal{C}}= \prod _{n} \operatorname{\mathcal{C}}_{n}$, and let $S: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be the shift functor (given on objects by the construction $\{ D_ n \} _{n \geq 0} \mapsto \{ U_{n}( D_{n+1} ) \} _{n \geq 0}$). By virtue of Corollary 4.5.7.19, the diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \varprojlim \operatorname{\mathcal{C}}_{n} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ (\operatorname{id}, \operatorname{id}) } \\ \operatorname{\mathcal{C}}\ar [r]^-{ (\operatorname{id},S) } & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}} \]
is a categorical pullback square, in which the maps are given by $(\kappa ,\lambda )$-finitary functors. By virtue of Corollary 9.2.8.5, to show that the sequence $C = \{ C_ n \} _{n \geq 0}$ is $(\kappa ,\lambda )$-compact as an object of the inverse limit $\varprojlim (\operatorname{\mathcal{C}}_{n})$, it will suffice to show that it is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{C}}$ and that $(C,C)$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}$. Both assertions follow from Proposition 9.2.8.6 (since $\kappa $ is uncountable and each $C_ n$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{n}$).
$\square$