Kerodon

Remark 8.5.4.7. Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a collection of $\infty $-categories with product $\prod _{i \in I} \operatorname{\mathcal{C}}_{i}$. Then an idempotent in $\operatorname{\mathcal{C}}$ is split if and only if its image in each factor $\operatorname{\mathcal{C}}_{i}$ is split. In particular, if each of the $\infty $-categories $\operatorname{\mathcal{C}}_{i}$ is idempotent complete, then $\operatorname{\mathcal{C}}$ is idempotent complete.