Corollary 9.4.6.19. Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a small collection of accessible $\infty $-categories. Then the product $\prod _{i \in I} \operatorname{\mathcal{C}}_ i$ is also accessible.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Variant 9.4.6.18, we can choose a small regular cardinal $\lambda $ such $I$ is $\lambda $-small and each $\operatorname{\mathcal{C}}_ i$ is $\lambda $-accessible. In this case, Remark 9.4.6.4 implies that $\prod _{i \in I} \operatorname{\mathcal{C}}_ i$ is $\lambda $-accessible. $\square$