Kerodon

Proof. Let $\{ S_ i \} _{i \in I}$ be a collection of $\kappa $-small simplicial sets indexed by a finite set $I$; we wish to show that the product $S = {\prod }_{i \in I} S_ i$ is $\kappa $-small. Without loss of generality, we may assume that $\kappa $ is the least infinite cardinal for which each of the simplicial sets $S_{i}$ is $\kappa $-small. Then $\kappa $ is regular (Remark 4.7.4.7). If $\kappa = \aleph _0$, then the desired result follows from Remark 3.6.1.6. We may therefore assume that $\kappa $ is uncountable. In this case, the desired result follows from the criterion of Proposition 4.7.4.10, since the collection of $\kappa $-small sets is closed under finite products (Proposition 4.7.3.5). $\square$