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Example 7.6.4.4. A (strictly) commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_{1} \ar [r] & \operatorname{\mathcal{C}}} \]

is a categorical pullback square (in the sense of Definition 4.5.2.7) if and only if the induced diagram $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{QCat}) = \operatorname{\mathcal{QC}}$ is a pullback square in the $\infty $-category $\operatorname{\mathcal{QC}}$. This follows by combining Corollaries 7.5.5.8 and 7.5.5.10.