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Corollary 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 the vertical maps are weak homotopy equivalences. Then the induced map

\[ A_0 {\coprod }^{\mathrm{h}}_{A} A_1 \rightarrow B_0 {\coprod }^{\mathrm{h}}_{B} B_1 \]

is also a weak homotopy equivalence.

Proof. Apply Corollary to the diagram

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times A \ar [d] & \operatorname{\partial \Delta }^1 \times A \ar [l] \ar [r] \ar [d] & A_0 \coprod A_1 \ar [d] \\ \Delta ^1 \times B & \operatorname{\partial \Delta }^1 \times B \ar [l] \ar [r] & B_0 \coprod B_1. } \]