Example 7.6.3.3. A (strictly) commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_{1} \ar [r] & A_{01} } \]
is a homotopy pushout square (in the sense of Definition 3.4.2.1) if and only if the induced diagram $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan}) = \operatorname{\mathcal{S}}$ is a pushout square in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$. This follows by combining Corollaries 7.5.7.7 and 7.5.7.9.