Kerodon

Proof. Fix a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ admit small $\kappa $-filtered colimits which are preserved by the functors $F_{-}$ and $F_{+}$. Using Variant 9.4.7.15, we can choose a small regular cardinal $\lambda > \kappa $ such that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$ and $\operatorname{\mathcal{C}}$ are $\lambda $-accessible and the functors $F_{-}$ and $F_{+}$ are $\lambda $-compact. Apply Corollary 9.4.4.10, we deduce that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $\lambda $-compactly generated. Let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $\lambda $-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 9.4.4.10, we can identify $\operatorname{\mathcal{C}}_{\pm }^{0}$ with the homotopy fiber product $\operatorname{\mathcal{C}}_{-}^{0} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}^{0} } \operatorname{\mathcal{C}}_{+}^{0}$, so the desired result is a special case of Corollary 4.9.5.18. $\square$