Remark 5.1.2.8. Assuming Theorem 5.1.2.2, one can give a more direct proof of Lemma 5.1.2.7. Let $q: X \rightarrow S$ be a left fibration of simplicial sets with the property that, for every vertex $s \in S$, the fiber $X_{s}$ is a contractible Kan complex. For each edge $e: s \rightarrow s'$ of $S$, the covariant transport $e_{!}: X_{s} \rightarrow X_{s'}$ is a morphism between contractible Kan complexes, and is therefore automatically a homotopy equivalence. Applying Theorem 5.1.2.2, we deduce that $q$ is a Kan fibration. Since the fibers of $q$ are contractible, Proposition 3.3.7.4 guarantees that $q$ is a trivial Kan fibration.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$