Remark Assuming Theorem, one can give a more direct proof of Lemma 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, we deduce that $q$ is a Kan fibration. Since the fibers of $q$ are contractible, Proposition guarantees that $q$ is a trivial Kan fibration.