Proposition 9.4.8.4 (Accessibility of Oriented Fiber Products). Let $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ be accessible $\infty $-categories and suppose we are given accessible functors $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$. Then the oriented fiber product $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is accessible.
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Proof. Using Variant 9.4.7.15, we can choose a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$ and $\operatorname{\mathcal{C}}$ are $\kappa $-accessible and the functors $F_{-}$ and $F_{+}$ are $\kappa $-compact. Applying Corollary 9.4.3.6, we deduce that $\operatorname{\mathcal{C}}_{\pm }$ is $\kappa $-compactly generated. Let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $\kappa $-compact objects, and define $\operatorname{\mathcal{C}}^{0}_{-}$, $\operatorname{\mathcal{C}}_{+}^{0}$, and $\operatorname{\mathcal{C}}^{0}_{\pm }$ similarly. We will complete the proof by showing that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }^{0}$ is essentially small. By assumption, the $\infty $-categories $\operatorname{\mathcal{C}}_{-}^{0}$, $\operatorname{\mathcal{C}}_{+}^{0}$, and $\operatorname{\mathcal{C}}^{0}$ are essentially small. Using Corollary 9.4.3.6, we can identify $\operatorname{\mathcal{C}}_{\pm }^{0}$ with the oriented fiber product $\operatorname{\mathcal{C}}_{-}^{0} \vec{\times }_{ \operatorname{\mathcal{C}}^{0} } \operatorname{\mathcal{C}}_{+}^{0}$, so the desired result is a special case of Proposition 4.9.5.17. $\square$