Proposition 9.2.8.3 (Compactness in Oriented Fiber Products). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ be $\infty $-categories which are $(\kappa ,\lambda )$-cocomplete, and let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ be functors which are $(\kappa ,\lambda )$-finitary. Then:
- $(1)$
The oriented fiber product $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is $(\kappa ,\lambda )$-cocomplete, and the projection maps $\operatorname{\mathcal{C}}_{-} \xleftarrow {\operatorname{ev}_{-}} \operatorname{\mathcal{C}}_{\pm } \xrightarrow {\operatorname{ev}_{+}} \operatorname{\mathcal{C}}_{+}$ are $(\kappa ,\lambda )$-finitary.
- $(2)$
Let $C$ be an object of $\operatorname{\mathcal{C}}_{\pm }$, given by a triple $(C_{-}, C_{+}, u)$ where $C_{-}$ is an object of $\operatorname{\mathcal{C}}_{-}$, $C_{+}$ is an object of $\operatorname{\mathcal{C}}_{+}$, and $u: F_{-}(C_{-} ) \rightarrow F_{+}( C_{+} )$ is a morphism in $\operatorname{\mathcal{C}}$. If $C_{-}$, $C_{+}$, and $F_{-}(C_{-} )$ are $(\kappa ,\lambda )$-compact (as objects of $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$, respectively), then $C$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{\pm }$.
Proof of Proposition 9.2.8.3.
Assertion $(1)$ follows from Corollary 7.1.9.5 (and does not require the assumption that $F_{+}$ is $(\kappa ,\lambda )$-finitary). We will prove $(2)$. Fix an uncountable cardinal $\mu $ of cofinality $\geq \lambda $ such that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ are locally $\mu $-small. We must show that the corepresentable functor $h^{C}: \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is $(\kappa ,\lambda )$-finitary. Let $h^{C_{-}}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$, $h^{C_{+}}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ and $h^{ F_{-}(C_{-} ) }: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ denote functors corepresented by $C_{-}$, $C_{+}$, and $F_{-}(C_{-})$, respectively. Let $\operatorname{ev}_{-}: \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ and $\operatorname{ev}_{+}: \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{+}$ be the projection maps. Applying Theorem 8.3.7.1 (and Remark 8.3.7.2), we obtain a (levelwise) pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ h^{C} \ar [r] \ar [d] & h^{C_{+}} \circ \operatorname{ev}_{+} \ar [d] \\ h^{C_{-}} \circ \operatorname{ev}_{-} \ar [r] & h^{ F_{-}(C_{-} )} \circ F_{+} \circ \operatorname{ev}_{+} } \]
in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}_{< \mu } )$. Since the collection of $(\kappa ,\lambda )$-finitary functors from $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{S}}_{< \mu }$ is closed under $\kappa $-small limits, it will suffice to show that the functors $h^{C_{+}} \circ \operatorname{ev}_{+}$, $h^{C_{-}} \circ \operatorname{ev}_{-}$, and $h^{ F_{-}(C_{-} )} \circ F_{+} \circ \operatorname{ev}_{+}$ are $(\kappa ,\lambda )$-finitary. Since the functors $\operatorname{ev}_{-}$, $\operatorname{ev}_{+}$, and $F_{+}$ are $(\kappa ,\lambda )$-finitary (Corollary 7.1.9.5), this follows from our assumption that the objects $C_{-}$, $C_{+}$, and $F_{-}( C_{-} )$ are $(\kappa ,\lambda )$-compact.
$\square$