Corollary 4.7.3.6. Let $\kappa $ be an infinite cardinal. Then the collection of $\kappa $-small sets is closed under finite coproducts.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\{ S_ i \} _{i \in I}$ be a finite collection of $\kappa $-small sets. Then the disjoint union ${\coprod }_{ i \in I} S_ i$ can be identified with a subset of the product ${\prod }_{i \in I} (S_ i \coprod \{ i \} )$, which is $\kappa $-small by virtue of Proposition 4.7.3.5. $\square$