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Definition 4.5.2.7. A commutative diagram of $\infty $-categories

4.21
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}

is a categorical pullback square if the composite map

\[ \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1} \hookrightarrow \operatorname{\mathcal{C}}_{0} \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1 \]

is an equivalence of $\infty $-categories.