In chapter 9 of Diamond-Shurman's book A First Course in Modular Forms, when they construct the Tate module associated to the modular curve $X_1(N)$, they state that the Galois action and Hecke action on the Picard group $\mathrm{Pic}^0(X_1(N))$ commute (on page 387).
While I understand why Hecke operators commute with Hecke operators, why would a Hecke operator commute with an element of the Galois group? In other words, for $T\in\mathbb{T}_{\mathbb{Z}}$ and $\sigma\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, why does \begin{equation} \sigma(T(x))=T(\sigma(x)) \end{equation} for all $x\in\mathrm{Pic}^0(X_1(N))$? A priori, the Galois group is noncommutative, so $\sigma\sigma'\ne\sigma'\sigma$ for all $\sigma,\sigma'\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. I feel like I'm missing something basic about group actions or (Galois) modules, but I can't see what.