The definition for affine schemes I have learned was that it is a locally ringed space $(X, O_X)$ that is isomorphic to $(Spec \ A, O_{ Spec \ A })$ for some ring $A$ (commutative and includes $1$). Then, it follows that $A \cong O_X(X)$. Therefore, it follows that $(X, O_X)$ is isomorphic to $(Spec \ O_X(X), O_{ Spec \ O_X(X) })$. Say this isomorphism is given by $(\phi, \alpha)$ where $\phi$ is a homeomorphism and $\alpha: O_{ Spec \ O_X(X)} \rightarrow \phi_* O_X$.
We know that the global section of $(Spec \ O_X(X), O_{ Spec \ O_X(X) })$ is $O_X(X)$, which is the same as that of $(X, O_X)$.
My question is, does it follow from the definition of affine schemes that the map between the global sections induced by $\alpha$ be necessarily the identity map? Thanks!