Kerodon

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Corollary 9.5.4.6. Suppose we are given a categorical pullback diagram of $\infty $-categories

9.16
\begin{equation} \begin{gathered}\label{equation:presentable-categorical-pullback} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm } \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{-} \ar [d]^{F_{-} } \\ \operatorname{\mathcal{C}}_{+} \ar [r]^-{F_{+} } & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}

where $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ are presentable. Assume either that $F_{-}$ and $F_{+}$ are both cocontinuous or both continuous and accessible. Then the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is presentable.

Proof. The assumption that (9.16) is a categorical pullback square guarantees that $\operatorname{\mathcal{C}}_{\pm }$ is equivalent to the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$, which is presentable by Remark 9.5.4.5. $\square$