Definition 3.4.2.1. 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 homotopy pushout square if, for every Kan complex $X$, the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A_{01}, X) \ar [r] \ar [d] & \operatorname{Fun}(A_{0}, X) \ar [d] \\ \operatorname{Fun}(A_1, X) \ar [r] & \operatorname{Fun}(A, X) } \]
is homotopy pullback square (Definition 3.4.1.1).