Corollary 4.8.8.14. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr] & & Y \ar [dl] \\ & S & } \]
with the following property: for every simplex $\sigma : \Delta ^ k \rightarrow S$, the induced map $f_{\sigma }: \Delta ^ k \times _{S} X \rightarrow \Delta ^ k \times _{S} Y$ is a categorical equivalence of simplicial sets. Then $f$ is a categorical equivalence of simplicial sets.