Remark 7.6.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative diagram in $\operatorname{\mathcal{C}}$. Then $\sigma $ is a pushout diagram in $\operatorname{\mathcal{C}}$ if and only if the opposite diagram $\sigma ^{\operatorname{op}}: \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a pullback diagram in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$; here we implicitly identify the simplicial set $\Delta ^1 \times \Delta ^1$ with its opposite (beware that there are two possible identifications we could choose, but the choice does not matter by virtue of Remark 7.6.2.8).
More generally, if $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories, then $\sigma $ is a $U$-pushout diagram if and only if $\sigma ^{\operatorname{op}}$ is a $U^{\operatorname{op}}$-pullback diagram.