On page 85 of Jech's Set Theory (3rd Edition), a complete Boolean algebra $B$ is defined to be $\kappa$-distributive if
\begin{equation}\label{a}\tag{1} \prod_{\alpha < \kappa}\, \sum_{i \in I_\alpha} u_{\alpha, i} = \sum_{f \in \prod_{\alpha < \kappa}} \prod_{\alpha < \kappa} u_{\alpha, f(\alpha)} \end{equation}
for any collection $\{u_{\alpha, i}\}_{\alpha < \kappa, i \in I_\alpha}$ of $B$. This property is easily seen to hold for a complete algebra of sets; for an arbitrary Boolean algebra, Jech presents two different characterizations of \eqref{a} in Lemma 7.16.
My question is why $\kappa$-distributivity is not just a consequence of Stone's Representation Theorem (Theorem 7.11): Suppose I wish to verify \eqref{a} for some Boolean algebra $B$. By Stone's Representation Theorem, there is an isomorphism $\varphi$ mapping $B$ to $S$, where $S$ is an algebra of sets. It suffices to verify that
\begin{equation} \varphi\left(\prod_{\alpha < \kappa}\, \sum_{i \in I_\alpha} u_{\alpha, i}\right) = \varphi\left(\sum_{f \in \prod_{\alpha < \kappa}} \prod_{\alpha < \kappa} u_{\alpha, f(\alpha)}\right). \end{equation}
To that end,
\begin{align} \varphi\left(\prod_{\alpha < \kappa}\, \sum_{i \in I_\alpha} u_{\alpha, i}\right) &= \bigcap_{\alpha < \kappa} \varphi\left(\sum_{i \in I_\alpha} u_{\alpha, i}\right)\tag{2}\\ &= \bigcap_{\alpha < \kappa}\,\bigcup_{i \in I_\alpha} \varphi\left(u_{\alpha, i}\right)\tag{3}\\ &= \bigcup_{f \in \prod_{\alpha < \kappa}} \bigcap_{\alpha < \kappa} \varphi\left(u_{\alpha, f(\alpha)}\right)\tag{4}\\ &\:\,\, \vdots\\ &= \varphi\left(\sum_{f \in \prod_{\alpha < \kappa}} \prod_{\alpha < \kappa} u_{\alpha, f(\alpha)}\right)\tag{5}, \end{align}
where (2), (3) and (5) hold because an isomorphism is a complete homomorphism; and where (4) is due to \eqref{a} and $S$ being an algebra of sets.
Certainly this argument is flawed, but I cannot pinpoint where I go wrong.
