Corollary 7.6.3.24. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative square, represented informally by the diagram
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r]^-{f'} \ar [d] & Y' \ar [d] \\ X \ar [r]^-{f} & Y. } \]
Then:
- $(1)$
If $f$ is an isomorphism, then $\sigma $ is a pullback square if and only if $f'$ is also an isomorphism.
- $(2)$
If $f'$ is an isomorphism, then $\sigma $ is a pushout square if and only if $f$ is also an isomorphism.