Let $\cal C, \cal D$ be model categories. Hovey says in his monograph "Model Categories" that the homotopy category $\operatorname{Ho}(\cal C \times D)$ is isomorphic to $\operatorname{Ho}(\cal C) \times \operatorname{Ho} (\cal D)$, and that this is true for any (finite I assume) number of model categories.

Is this true in general? Let $\{ \mathcal C_i \}_{i \in I}$ be a family of categories, let $W_i \subseteq \operatorname{Mor} \cal C_i$ be families of morphisms and let $\mathcal C = \prod_{i \in I} \cal C_i$. Are the localisations $$\mathcal C\left[ \prod_{i \in I} W_i ^{-1}\right] \cong \prod_{i \in I} \mathcal C_i\left[W_i^{-1}\right]?$$

I can find a functor from the left to the right in general.

Is it at least true for infinite families of model categories?