Corollary 4.7.4.14. Let $\lambda $ be an infinite cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be its exponential cofinality (Definition 4.7.3.16). Then the collection of $\lambda $-small simplicial sets is closed under $\kappa $-small products.
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Proof. Let $\{ S_ i \} _{i \in I}$ be a collection of $\lambda $-small simplicial sets indexed by a $\kappa $-small set $I$; we wish to show that the product $S = {\prod }_{i \in I} S_ i$ is $\lambda $-small. If $\kappa = \aleph _0$, this follows from Corollary 4.7.4.11. We may therefore assume that $\kappa $ is uncountable. Then the cofinality $\mathrm{cf}(\lambda )$ is also uncountable (Remark 4.7.3.17). The desired result now follows from the criterion of Proposition 4.7.4.9, since the collection of $\lambda $-small sets is closed under $\kappa $-small products. $\square$