Kerodon

Remark 9.5.4.5. In the situation of Proposition 9.5.4.4, the homotopy fiber product $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is accessible (Proposition 9.4.8.9). If the functors $F_{-}$ and $F_{+}$ are both cocontinuous, then $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is closed under small colimits in the presentable $\infty $-category $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ (Remark 7.1.9.6), and is therefore also presentable. If $F_{-}$ and $F_{+}$ are both continuous, then $\operatorname{\mathcal{C}}_{\pm }$ is closed under small limits in $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$, and therefore also presentable.