Kerodon

Proof. Suppose that $\kappa $ is not regular: that is, there is a collection of $\kappa $-small sets $\{ T_{s} \} _{s \in S}$ indexed by a $\kappa $-small set $S$ such that $T = {\coprod }_{s \in S} T_ s$ has cardinality $\geq \kappa $. Choose a collection of $\lambda $-small sets $\{ U_{t} \} _{t \in T}$ for which the product $U = {\prod }_{t \in T} U_ t$ is not $\lambda $-small. For each $s \in S$, let $U_{s}$ denote the product ${\prod }_{ t \in T_{s} } U_{t}$. Since $T_{s}$ is $\mathrm{ecf}(\lambda )$-small, the set $U_{s}$ is $\lambda $-small. Since $S$ is also $\mathrm{ecf}(\lambda )$-small, it follows that $U \simeq {\prod }_{s \in S} U_ s$ is also $\lambda $-small, which is a contradiction. $\square$