Warning 11.5.0.53. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ C \ar [d] & A \ar [d] \ar [l] \ar [r] & B \ar [d] \\ C' & A' \ar [l] \ar [r] & B' } \]
in which the vertical maps are weak homotopy equivalences. Then the induced map $C \coprod _{A} B \rightarrow C' \coprod _{A'} B'$ need not be a weak homotopy equivalence. For example, the pushout of the upper half of the diagram
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \ar [d] & \operatorname{\partial \Delta }^{1} \ar [l] \ar [r] \ar@ {=}[d] & \Delta ^1 \ar [d] \\ \Delta ^0 & \operatorname{\partial \Delta }^{1} \ar [l] \ar [r] & \Delta ^0 } \]
is not weakly contractible (it has nontrivial homology in degree $1$), but the pushout of the lower half is isomorphic to $\Delta ^0$.