\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& & & \mathbb{G}_{m}\\
E^{*} \arrow[r] & \left(\prod_{E/\mathbb{Q}} \mathbb{G}_{m}\right)\big/U \times I(E)/I^{\circ}(E) \arrow[rr] & & S_{E} \arrow[u]\\
E^{*} \arrow[r] \arrow[u, no head] & I(E) \times I(E)/I^{\circ}(E) \arrow[rr] & & S_{E}(\mathbb{A})\\
E^{*} \arrow[rr] \arrow[u, no head] & & I(E) \arrow[r] & C(E) \arrow[u]
\end{tikzcd}