Kerodon

Variant 4.7.5.9. Let $\kappa $ be an uncountable cardinal and let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a collection of essentially $\kappa $-small $\infty $-categories. Suppose that the cardinality of the index set $I$ has smaller than the exponential cofinality $\mathrm{ecf}(\kappa )$. Then the product ${\prod }_{i \in I} \operatorname{\mathcal{C}}_{i}$ is also essentially $\kappa $-small. This follows by combining Corollary 4.7.4.14 with Remark 4.5.1.17.