Corollary 3.3.7.5. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ Y \ar [rr]^{u} \ar [dr] & & X \ar [dl] \\ & S, & } \]
where the vertical maps are Kan fibrations. Then $u$ is a weak homotopy equivalence if and only if every vertex $s \in S$ satisfies the following condition:
- $(\ast _ s)$
The induced map of fibers $u_{s}: X_{s} \rightarrow Y_{s}$ is a homotopy equivalence of Kan complexes.