Proposition 9.4.4.11. Let $\kappa \trianglelefteq \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 the following conditions:
The $\infty $-categories $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $(\kappa ,\lambda )$-compactly generated and the $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-cocomplete.
The functor $F_{-}$ and $F_{+}$ are $(\kappa ,\lambda )$-compact.
There exists a regular cardinal $\kappa _0 < \kappa $ for which the full subcategories of $(\kappa ,\lambda )$-compact objects $\operatorname{\mathcal{C}}^{0}_{-} \subseteq \operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}^{0}_{+} \subseteq \operatorname{\mathcal{C}}_{+}$ admits $\kappa _0$-sequential colimits, which are preserved by the functors $F_{-}$ and $F_{+}$.
Then the homotopy fiber product $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is $(\kappa ,\lambda ))$-compactly generated. Moreover, an object of $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda ))$-compact if and only if its images in $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $(\kappa ,\lambda ))$-compact.
Proof.
Let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the smallest full subcategory which contains $\operatorname{\mathcal{C}}^0$ and is closed under $\lambda $-small $\kappa $-filtered colimits. Then $\operatorname{\mathcal{C}}'$ is $(\kappa ,\lambda )$-compactly generated and our hypotheses guarantee that the functors $F_{-}$ and $F_{+}$ factor through $\operatorname{\mathcal{C}}'$. We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$ and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is also $(\kappa ,\lambda )$-compactly generated.
Fix a regular cardinal $\mu $ satisfying $\lambda \trianglelefteq \mu $ such that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ are essentially $\mu $-small. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$ and $\operatorname{\mathcal{C}}$ are given as full subcategories of their $\operatorname{Ind}_{\lambda }^{\mu }$-completions $\widehat{\operatorname{\mathcal{C}}}_{-} = \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{\mathcal{C}}_{-} )$, $\widehat{\operatorname{\mathcal{C}}}_{+} = \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{\mathcal{C}}_{+}$, and $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{\mathcal{C}})$, and that $F_{-}$ and $F_{+}$ are the restrictions of $(\lambda ,\mu )$-compact functors $\widehat{F}_{-}: \widehat{\operatorname{\mathcal{C}}}_{-} \rightarrow \widehat{\operatorname{\mathcal{C}}}$ and $\widehat{F}_{+}: \widehat{\operatorname{\mathcal{C}}}_{+} \rightarrow \widehat{\operatorname{\mathcal{C}}}$. Applying Propositions 9.4.1.25, we see that the $\infty $-categories $\widehat{\operatorname{\mathcal{C}}}_{-}$, $\widehat{\operatorname{\mathcal{C}}}_{+}$, and $\widehat{\operatorname{\mathcal{C}}}$ are $(\kappa ,\mu )$-compactly generated, and that their full subcategories of $(\kappa ,\mu )$-compact objects can be identified with $\operatorname{\mathcal{C}}_{-}^{0}$, $\operatorname{\mathcal{C}}_{+}^{0}$ and $\operatorname{\mathcal{C}}^{0}$, respectively. Applying Corollary 9.4.4.10 for the pairs $(\lambda \leq \mu )$ and $(\kappa \leq \mu )$, we obtain the following:
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}_{\pm } = \widehat{\operatorname{\mathcal{C}}}_{-} \times ^{\mathrm{h}}_{ \widehat{\operatorname{\mathcal{C}}} } \widehat{\operatorname{\mathcal{C}}}_{+}$ is $(\lambda ,\mu )$-compactly generated. Moreover, an object of $\widehat{\operatorname{\mathcal{C}}}_{\pm }$ is $(\lambda ,\mu )$-compact if and only if it belongs to the full subcategory $\operatorname{\mathcal{C}}_{\pm }$.
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}_{\pm }$ is $(\kappa ,\mu )$-compactly generated. Moreover, an object of $\widehat{\operatorname{\mathcal{C}}}_{\pm }$ is $(\kappa ,\mu )$-compact if and only if it belongs to the full subcategory $\operatorname{\mathcal{C}}^{0}_{\pm } = \operatorname{\mathcal{C}}^{0}_{-} \times _{\operatorname{\mathcal{C}}^{0}}^{\mathrm{h}} \operatorname{\mathcal{C}}^{0}_{+}$.
Applying Corollary 9.4.1.26, we see that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-compactly generated, and that an object of $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-compact if and only if it belongs to the full subcategory $\operatorname{\mathcal{C}}^{0}_{\pm }$.
$\square$