I'm a little bit confused on a construction (2.2) in Definition 2.1 from Mumford's & Oda's Algebraic Geometry II on pages 124-125
let $k$ a field and $\bar{k}$ the algebraic closure of $k$. let $Gal(\bar{k} / k)$ the Galois group. then every $\sigma_k \in Gal(\bar{k} / k)$ induces a map $\sigma_k: Spec \text{ } \bar{k} \to Spec \text{ } \bar{k}$ that is defined by $(\sigma_k)^*a= \sigma^{-1}a$ for $a \in \bar{k}$. let $\overline{X}:= X \times_k \bar{k}$.
Definition 2.1 For a $k$-scheme X, define the conjugation action of $Gal(\bar{k} / k)$ on $\overline{X}$ to be:
$$\sigma_X = 1_X \times \sigma_k : \overline{X} \to \overline{X}, \text{ all } \sigma_k \in Gal(\bar{k} / k)$$
Then $\sigma_X$ fits into a diagram:
$\require{AMScd}$ \begin{CD} \overline{X} @>{\sigma_X}>> \overline{X}\\ @VVV @VVV\\ Spec \text{ } \bar{k} @>{\sigma_k}>> Spec \text{ } \bar{k} \end{CD}
What this means is that if $f \in \mathcal{O}_{\overline{X}}(U)$ then $\sigma_X^*f \in \mathcal{O}_{\overline{X}}(\sigma_X ^{-1} U)$ has value at a point $x \in \sigma_X ^{-1} U$ given by:
\begin{equation} (\sigma_X^*f)(x)=\sigma^{-1} \cdot f(\sigma_X \cdot x) \tag{2.2}\label{eq:0105star} \end{equation}
(???)
Q: I don't understand why the commutativity of diagram above imply that the equation for the sections (2.2) is true.
$\sigma_X$ induced maps $\sigma_X^*: \mathcal{O}_{\overline{X}}( U) \to \mathcal{O}_{\overline{X}}(\sigma_X ^{-1} U)$ and $(\sigma_X^*)_x: \mathcal{O}_{\overline{X}, \sigma_X \cdot x} \to \mathcal{O}_{\overline{X},x}$. evaluation of a section $f \in \mathcal{O}_{\overline{X}}( U)$ is nothing but composition $\mathcal{O}_{\overline{X}}( U) \to \mathcal{O}_{\overline{X},x} \to \mathbb{k}(x)$ of canonical maps. why these observatons imply (2.2)? is the map $ \mathbb{k}(\sigma_X \cdot x) \to \mathbb{k}(x) $ fully determined by $\sigma_k$ from the diagram?