Definition 4.5.4.1. A commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]
is a categorical pushout square if, for every $\infty $-category $\operatorname{\mathcal{C}}$, the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,\operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}(A_0,\operatorname{\mathcal{C}})^{\simeq } \ar [l] \\ \operatorname{Fun}(A_1, \operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}(A_{01}, \operatorname{\mathcal{C}})^{\simeq } \ar [u] \ar [l] } \]
is a homotopy pullback square.