Corollary 11.10.1.21. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^-{f} \ar [dr]_{q} & & X' \ar [dl]^{q'} \\ & S, & } \]
where $q$ and $q'$ are left fibrations. The following conditions are equivalent:
The morphism $f$ is an equivalence of left fibrations over $S$.
For every vertex $s \in S$, the induced map of Kan complexes $f_{s}: X_{s} \rightarrow X'_{s}$ is a homotopy equivalence.