Corollary 3.4.2.14. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A_0 \ar [d] & A \ar [l]_{f_0} \ar [r]^-{f_1} \ar [d] & A_1 \ar [d] \\ B_0 & B \ar [l]_{g_0} \ar [r]^-{g_1} & B_1, } \]
where $f_0$ and $g_0$ are monomorphisms and the vertical maps are weak homotopy equivalences. Then the induced map
\[ A_0 \coprod _{A} A_1 \rightarrow B_0 \coprod _{B} B_1 \]
is a weak homotopy equivalence.