Remark 4.7.5.7 (Coproducts). Let $\kappa $ be an uncountable cardinal and let $\{ \operatorname{\mathcal{C}}_{i} \} _{i \in I}$ be a collection of essentially $\kappa $-small simplicial sets. Suppose that the cardinality of the index set $I$ is smaller than the cofinality $\mathrm{cf}(\kappa )$. Then the coproduct ${\coprod }_{i \in I} \operatorname{\mathcal{C}}_ i$ is also essentially $\kappa $-small. This follows by combining Remark 4.7.4.5 with Corollary 4.5.3.10. In particular:
The collection of essentially $\kappa $-small simplicial sets is closed under finite coproducts.
If $\kappa $ is regular, then the collection of essentially $\kappa $-small simplicial sets is closed under $\kappa $-small coproducts.