Remark 5.4.4.5 (Coproducts). Let $\kappa $ be an infinite cardinal and let $\{ S_{i} \} _{i \in I}$ be a collection of $\kappa $-small simplicial sets. Suppose that the cardinality of the index set $I$ is smaller than the cofinality $\mathrm{cf}(\kappa )$. Then the coproduct $\coprod _{i \in I} S_ i$ is also $\kappa $-small (see Corollary 5.4.3.9). In particular:

The collection of $\kappa $-small simplicial sets is closed under finite coproducts.

If $\kappa $ is regular, then the collection of $\kappa $-small simplicial sets is closed under $\kappa $-small coproducts.