Corollary 3.3.7.3. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { Y \ar [rr]^{u} \ar [dr] & & X \ar [dl] \\ & S, & } \]

where the vertical maps are Kan fibrations. Then $u$ is a weak homotopy equivalence if and only if, for each vertex $s \in S$, the induced map $u_{s}: Y_{s} \rightarrow X_{s}$ is a homotopy equivalence of Kan complexes.