Example 7.6.2.2. Let $\operatorname{\mathcal{C}}$ be an ordinary category. Then diagram $\sigma : [1] \times [1] \rightarrow \operatorname{\mathcal{C}}$ is a pullback square in $\operatorname{\mathcal{C}}$ (in the sense of classical category theory) if and only if the induced map
\[ \operatorname{N}_{\bullet }(\sigma ): \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \]
is a pullback square in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 7.6.2.1); this follows from Example 7.1.1.4 and Remark 7.1.3.6. Similarly, $\sigma $ is a pushout square in $\operatorname{\mathcal{C}}$ if and only if $\operatorname{N}_{\bullet }(\sigma )$ is a pushout square in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.