Corollary 4.7.4.13. Let $\kappa $ be an infinite cardinal and let $K$ and $L$ be $\kappa $-small simplicial sets. Then the join $K \star L$ is $\kappa $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The case $\kappa = \aleph _0$ follows from Remark 4.3.3.21. We may therefore assume that $\kappa $ is uncountable. In this case, it will suffice to show that for every $n \geq 0$, the collection of $n$-simplices of $K \star L$ is $\kappa $-small (Proposition 4.7.4.10). This follows from Remark 4.3.3.17, since the collection of $\kappa $-small sets is closed under finite products and coproducts (Proposition 4.7.3.5 and Corollary 4.7.3.6). $\square$