Proposition 3.4.2.8 (Transitivity). Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [r] \ar [d] & C \ar [d] \\ A' \ar [r] & B' \ar [r] & C', } \]
where the left half is a homotopy pushout square. Then the right half is a homotopy pushout square if and only if the outer rectangle is a homotopy pushout square.