Corollary 9.4.4.10. We Let $\kappa \leq \lambda $ be regular cardinals and let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories satisfying conditions $(a)$, $(b)$, and $(c)$ of Theorem 9.4.4.9. Then the homotopy fiber product Then the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is $(\kappa ,\lambda )$-compactly generated. Moreover, an object of $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is $(\kappa ,\lambda )$-compact if and only if its images in $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $(\kappa ,\lambda ))$-compact.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Corollary 4.5.3.24 (and Corollary 4.5.3.21), we can reduce to the case where $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ is an isofibration. In this case, the inclusion map $\operatorname{\mathcal{C}}_{-} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \hookrightarrow \operatorname{\mathcal{C}}_{-} \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_{+}$ is an equivalence of $\infty $-categories (Corollary 4.5.3.29). The desired result now follows by applying Theorem 9.4.4.9 to the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{-} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{+} \ar [d]^{ F_{+} } \\ \operatorname{\mathcal{C}}_{-} \ar [r]^-{ F_{-} } & \operatorname{\mathcal{C}}, } \]
which is a categorical pullback square by virtue of Corollary 4.5.3.28. $\square$