Remark 7.6.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma ,\sigma ': \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be square diagrams which are isomorphic (when viewed as objects of the $\infty $-category $\operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{C}})$). Then $\sigma $ is a pullback square if and only if $\sigma '$ is a pullback square, and a pushout square if and only if $\sigma '$ is a pushout square.
More generally, if $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories, then then $\sigma $ is a $U$-pullback square if and only if $\sigma '$ is a $U$-pullback square, and $\sigma $ is a $U$-pushout square if and only if $\sigma '$ is a $U$-pushout square (see Proposition 7.1.6.15).