Lemma Let $q: X \rightarrow S$ be a left fibration of simplicial sets, where $S$ is a Kan complex. Suppose that $q$ induces a surjection $\pi _0(q): \pi _0(X) \rightarrow \pi _0(S)$. Then $q$ is surjective on vertices.

Proof. Fix a vertex $s \in S$. Since $\pi _0(q)$ is surjective, there exists a vertex $x \in X$ for which $q(x)$ and $s$ belong to the same connected component of $\pi _0(S)$. Since $S$ is a Kan complex, we can choose an edge $e: q(x) \rightarrow s$ in the simplicial set $S$. Our assumption that $q$ is a left fibration guarantees that we can write $e = q(\overline{e})$ for some edge $\overline{e}: x \rightarrow \overline{s}$ of the simplicial set $X$. In particular, there exists a vertex $\overline{s} \in X$ satisfying $q( \overline{s} ) = s$. $\square$