$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 9.2.8.5 (Compactness in Homotopy Fiber Products). Let $\kappa \leq \lambda $ be regular cardinals and suppose we are given a categorical pullback diagram of $\infty $-categories
9.12
\begin{equation} \begin{gathered}\label{equation:compact-in-categorical-pullback} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm } \ar [r]^-{ E_{+} } \ar [d]^{ E_{-}} & \operatorname{\mathcal{C}}_{+} \ar [d]^{F_{+}} \\ \operatorname{\mathcal{C}}_{-} \ar [r]^-{F_{-}} & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
Assume that the $\infty $-categories $\operatorname{\mathcal{C}}_{+}$, $\operatorname{\mathcal{C}}_{-}$, and $\operatorname{\mathcal{C}}$ are $(\kappa ,\lambda )$-cocomplete and that the functors $F_{-}$ and $F_{+}$ are $(\kappa ,\lambda )$-finitary (so that $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-cocomplete and the functors $E_{+}$ and $E_{-}$ are $(\kappa ,\lambda )$-finitary; see Proposition 9.1.9.8). Let $C \in \operatorname{\mathcal{C}}_{\pm }$ be an object whose images in $\operatorname{\mathcal{C}}_{+}$, $\operatorname{\mathcal{C}}_{-}$, and $\operatorname{\mathcal{C}}$ are $(\kappa ,\lambda )$-compact. Then $C$ is $(\kappa ,\lambda )$-compact.
Proof.
Our assumption that (9.12) is a categorical pullback square guarantees that $E_{-}$ and $E_{+}$ induce an equivalence of $\infty $-categories $E: \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$. It will therefore suffice to show that $E(C)$ is $(\kappa ,\lambda )$-compact when viewed as an object of the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$. In fact, $E(C)$ is even $(\kappa ,\lambda )$-compact when viewed as an object of the oriented fiber product $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$, by virtue of Proposition 9.2.8.3.
$\square$