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It is true for arbitrary products of model categories (or just cofibration categories) as proven in Theorem 7.1.1 of http://arxiv.org/abs/math/0610009v4.

It is also true for finite products of arbitrary relative categories as discussed in this answser: Localizing an arbitrary additive category. (For more details on this and related arguments see http://nforum.mathforge.org/discussion/4769/connected-components-preserve-finite-products/.)

Addendum: This doesn't hold for infinite products of arbitrary relative categories. Consider a sequence of relative categories $\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2, \ldots$ such that $\mathcal{C}_i$ is saturated and has objects $X_i$ and $Y_i$ that are weakly equivalent, but the shortest zig-zag of weak equivalences witnessing this is at least $i$ arrows long. Then $(X_0, X_1, \ldots)$ and $(Y_0, Y_1, \ldots)$ are isomorphic as objects of $\prod_i \mathrm{Ho}(\mathcal{C}_i)$ but not as objects of $\mathrm{Ho}(\prod_i \mathcal{C}_i)$ since this would imply that the length of the shortest zig-zag connecting $X_i$ to $Y_i$ is bounded.

It is true for arbitrary products of model categories (or just cofibration categories) as proven in Theorem 7.1.1 of http://arxiv.org/abs/math/0610009v4.

It is also true for finite products of arbitrary relative categories as discussed in this answser: Localizing an arbitrary additive category. (For more details on this and related arguments see http://nforum.mathforge.org/discussion/4769/connected-components-preserve-finite-products/.)

Addendum: This doesn't hold for infinite products of arbitrary relative categories. Consider a sequence of relative categories $\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2, \ldots$ such that $\mathcal{C}_i$ is saturated and has objects $X_i$ and $Y_i$ that are weakly equivalent, but the shortest zig-zag of weak equivalences witnessing this is at least $i$ arrows long. Then $(X_0, X_1, \ldots)$ and $(Y_0, Y_1, \ldots)$ are isomorphic as objects of $\prod_i \mathrm{Ho}(\mathcal{C}_i)$ but not as objects of $\mathrm{Ho}(\prod_i \mathcal{C}_i)$ since this would imply that the length of the shortest zig-zag connecting $X_i$ to $Y_i$ is bounded.

It is true for arbitrary products of model categories (or just cofibration categories) as proven in Theorem 7.1.1 of http://arxiv.org/abs/math/0610009v4.

It is also true for finite products of arbitrary relative categories as discussed in this answser: Localizing an arbitrary additive category. (For more details on this and related arguments see http://nforum.mathforge.org/discussion/4769/connected-components-preserve-finite-products/.)

Addendum: This doesn't hold for infinite products of arbitrary relative categories. Consider a sequence of relative categories $\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2, \ldots$ such that $\mathcal{C}_i$ has objects $X_i$ and $Y_i$ that are weakly equivalent, but the shortest zig-zag of weak equivalences witnessing this is at least $i$ arrows long. (We can safely assume that all these categories are saturated since passing to the saturation doesn't change the homotopy category.) Then $(X_0, X_1, \ldots)$ and $(Y_0, Y_1, \ldots)$ are isomorphic as objects of $\prod_i \mathrm{Ho}(\mathcal{C}_i)$ but not as objects of $\mathrm{Ho}(\prod_i \mathcal{C}_i)$ since this would imply that the length of the shortest zig-zag connecting $X_i$ to $Y_i$ is bounded.

It is true for arbitrary products of model categories (or just cofibration categories) as proven in Theorem 7.1.1 of http://arxiv.org/abs/math/0610009v4.

It is also true for finite products of arbitrary relative categories as discussed in this answser: Localizing an arbitrary additive category. (For more details on this and related arguments see http://nforum.mathforge.org/discussion/4769/connected-components-preserve-finite-products/.)

Addendum: This doesn't hold for infinite products of arbitrary relative categories. Consider a sequence of relative categories $\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2, \ldots$ such that $\mathcal{C}_i$ is saturated and has objects $X_i$ and $Y_i$ that are weakly equivalent, but the shortest zig-zag of weak equivalences witnessing this is at least $i$ arrows long. Then $(X_0, X_1, \ldots)$ and $(Y_0, Y_1, \ldots)$ are isomorphic as objects of $\prod_i \mathrm{Ho}(\mathcal{C}_i)$ but not as objects of $\mathrm{Ho}(\prod_i \mathcal{C}_i)$ since this would imply that the length of the shortest zig-zag connecting $X_i$ to $Y_i$ is bounded.

It is true for arbitrary products of model categories (or just cofibration categories) as proven in Theorem 7.1.1 of http://arxiv.org/abs/math/0610009v4.

It is also true for finite products of arbitrary relative categories as discussed in this answser: Localizing an arbitrary additive category. (For more details on this and related arguments see http://nforum.mathforge.org/discussion/4769/connected-components-preserve-finite-products/.)

It is true for arbitrary products of model categories (or just cofibration categories) as proven in Theorem 7.1.1 of http://arxiv.org/abs/math/0610009v4.

It is also true for finite products of arbitrary relative categories as discussed in this answser: Localizing an arbitrary additive category. (For more details on this and related arguments see http://nforum.mathforge.org/discussion/4769/connected-components-preserve-finite-products/.)

Addendum: This doesn't hold for infinite products of arbitrary relative categories. Consider a sequence of relative categories $\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2, \ldots$ such that $\mathcal{C}_i$ has objects $X_i$ and $Y_i$ that are weakly equivalent, but the shortest zig-zag of weak equivalences witnessing this is at least $i$ arrows long. (We can safely assume that all these categories are saturated since passing to the saturation doesn't change the homotopy category.) Then $(X_0, X_1, \ldots)$ and $(Y_0, Y_1, \ldots)$ are isomorphic as objects of $\prod_i \mathrm{Ho}(\mathcal{C}_i)$ but not as objects of $\mathrm{Ho}(\prod_i \mathcal{C}_i)$ since this would imply that the length of the shortest zig-zag connecting $X_i$ to $Y_i$ is bounded.