Proposition 9.5.4.4 (Presentability of Oriented Fiber Products). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be presentable $\infty $-categories and let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category. Suppose we are given a pair of accessible functors $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$. If $F_{-}$ is cocontinuous or $F_{+}$ is continuous, then the oriented fiber product $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is presentable.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It follows from Proposition 9.4.8.4 that the $\infty $-category $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is accessible. Applying Corollary 7.1.9.5, we see that it is either complete (in the case where $F_{+}$ is continuous) or cocomplete (in the case where $F_{-}$ is cocontinuous), and therefore presentable. $\square$