Variant 3.2.6.9. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between Kan complexes. Then the fiber $X_{s}$ is connected if and only if $f$ satisfies both of the following conditions:
The connected component $[s] \in \pi _0(S)$ has a unique preimage under the map $\pi _0(f): \pi _0(X) \rightarrow \pi _0(S)$ (given by $[x] \in \pi _0(X)$).
The map of fundamental groups $\pi _{1}(X,x) \rightarrow \pi _1(S,s)$ is surjective.