Remark 4.7.5.8 (Products). Let $\kappa $ be an uncountable cardinal and let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a finite collection of simplicial sets which are essentially $\kappa $-small. Then the product ${\prod }_{i \in I} \operatorname{\mathcal{C}}_ i$ is essentially $\kappa $-small. This follows by combining Corollary 4.7.4.18, since the collection of categorical equivalences is stable under the formation of finite products (Remark 4.5.3.7).
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