Kerodon

Corollary 9.4.4.6. Let $\kappa $ be a small regular cardinal, let $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ be $\kappa $-compactly generated $\infty $-categories, let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ be a $\kappa $-compact functor, and let $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ be a $\kappa $-finitary functor. Then the oriented fiber product $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is $\kappa $-compactly generated. Moreover, an object of $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is $\kappa $-compact if and only if its images in $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $\kappa $-compact.