Corollary 1.2.5.11. Let $\{ S_{\alpha } \} _{\alpha \in A}$ be a collection of Kan complexes parametrized by a set $A$, and let $S = \prod _{\alpha \in A} S_{\alpha \bullet }$ denote their product. Then the canonical map
\[ \pi _0( S ) \rightarrow \prod _{\alpha \in A} \pi _0 ( S_{\alpha } ) \]
is bijective. In particular, $S$ is connected if and only if each factor $S_{\alpha }$ is connected.