Galois equivariance of the diagonal embedding of global units

Let $K$ be a number field, $p$ a rational prime, $\mathcal P$ the set of primes of $\mathcal O_K$ above $p$, and $U=\prod_{v\in\mathcal P}\mathcal O_v^\times$ the group of semilocal units at $p$ with its product (profinite) topology. For $\sigma\in\operatorname{Aut}(K/\mathbb Q)$ let $\sigma_U:U\to U$ be the Galois action on semilocal units, $(\sigma_U x)_w=\sigma(x_{\sigma^{-1}w})$, where $\sigma:K_{\sigma^{-1}w}\xrightarrow{\sim}K_w$ is the continuous extension of $\sigma$ to completions (definition Def_LeopoldtGaloisAction).

Let $\iota:\mathcal O_K^\times\to U$ be the diagonal embedding of the global units. Then for every $\sigma\in\operatorname{Aut}(K/\mathbb Q)$ and every global unit $\varepsilon\in\mathcal O_K^\times$,

where $\sigma(\varepsilon)$ is the image of $\varepsilon$ under the restriction of $\sigma$ to $\mathcal O_K$.

That is, $\iota$ is $\operatorname{Aut}(K/\mathbb Q)$-equivariant; this is what makes the $p$-adic closure $\overline E$ of the global units a Galois-stable subgroup of $U$.