Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.7.3.8. Let $\lambda $ be an infinite cardinal. Then $\kappa = \mathrm{cf}(\lambda )$ is the smallest cardinal for which there exists a set $S$ of cardinality $\kappa $ and a collection of $\lambda $-small sets $\{ T_{s} \} _{s \in S}$, where the coproduct ${\coprod }_{s \in S} T_ s$ is not $\lambda $-small.

Proof. Proposition 4.7.2.14 guarantees that $\kappa $ is a cardinal. The characterization is a restatement of Proposition 4.7.3.7. $\square$