Remark 3.1.6.8. Let $\{ f_ i: X_ i \rightarrow Y_ i \} _{i \in I}$ be a collection of homotopy equivalences of simplicial sets indexed by a set $I$, and let $f: \prod _{i \in I} X_ i \rightarrow \prod _{i \in I} Y_ i$ be their product. Then:
If $I$ is finite, then $f$ is a homotopy equivalence. This follows from Remark 3.1.6.6 and Corollary 1.2.1.27.
If each of the simplicial sets $X_{i}$ and $Y_{i}$ is a Kan complex, then $f$ is a homotopy equivalence. This follows from Remark 3.1.6.6 and Corollary 1.2.5.11.
The morphism $f$ need not be a homotopy equivalence in general (see Warning 1.2.1.28).