Remark 7.7.5.2. In the situation of Definition 7.7.5.1, a cubical diagram can be regarded as a natural transformation from the top face to the bottom face, which is cartesian if and only if the remaining faces are pullback squares (see Proposition 7.6.2.28). It follows that:
The $\infty $-category $\operatorname{\mathcal{C}}$ satisfies the second Mather cube theorem if and only if every pushout square in $\operatorname{\mathcal{C}}$ is a universal pushout square.
If $\operatorname{\mathcal{C}}$ admits pushouts, then it satisfies both Mather cube theorems if and only if every pushout square in $\operatorname{\mathcal{C}}$ is a strongly universal pushout square (Corollary 7.7.2.19).