Kerodon

Remark 9.5.6.13 (Products). Let $\{ \operatorname{\mathcal{C}}(s) \} _{s \in S}$ be a collection of presentable $\infty $-categories indexed by a small set $S$, so that the product $\operatorname{\mathcal{C}}= \prod _{s \in S} \operatorname{\mathcal{C}}(s)$ is also presentable (Example 9.5.4.8). Suppose that, for each $s \in S$, we are given a Bousfield localization $\operatorname{\mathcal{C}}_0(s) \subseteq \operatorname{\mathcal{C}}(s)$. Then the product $\operatorname{\mathcal{C}}_0 = \prod _{s \in S} \operatorname{\mathcal{C}}_0(s)$ is a Bousfield localization of $\operatorname{\mathcal{C}}$.