Corollary 3.4.2.17. Let $f: X \rightarrow S$ be a morphism of simplicial sets. Suppose that, for every $k$-simplex $\Delta ^ k \rightarrow S$, the fiber product $\Delta ^{k} \times _{S} X$ is weakly contractible. Then $f$ is a weak homotopy equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Apply Proposition 3.4.2.16 in the special case $Y = S$. $\square$