Let $E=E^{sep}$ be a separably closed field, i.e. any separable polynomial over $E$ has a solution in $E$, and let $Var_E$ be the category of smooth, affine schemes of finite type over $E$.
On the other hand consider the full subcategory $Var_E^{classic}$ of locally ringed spaces $(X,\mathcal{O}_X)$, such that $X\subset\mathbb{A}^n(E)=E^n$ is a zariski-closed subspace. Then we have a functor $$Var_E\rightarrow Var_E^{classic}, X\mapsto (X(E),\mathcal{O}_{X(E)}),$$ where $X(E):=Mor_E(Spec(E),X)$ and the sheaf $\mathcal{O}_{X(E)}:=\alpha^{-1}\mathcal{O}_X$ is the inverse image for the canonical embedding $\alpha:X(E)\rightarrow X$.
My question is: Is this functor fully faithful?
For algebraically closed fields $E=E^{alg}$, this is true, since the canonical embedding $\alpha(X(E))\subset X$ is very dense (i.e. the inclusion $X(E)\subset X$ induces a bijection on the topologies) and so via sobrification of $X(E)$ (for any topological space Y only containing closed points, the sobrification $sob(Y)$ is a certain topological structure on the set of irreducible subsets of Y, such that the canonical embedding $Y\subset sob(Y)$ is continuous.), we obtain an inverse for morphisms $(X(E),\mathcal{O}_{X(E)})\rightarrow (Y(E),\mathcal{O}_{Y(E)})$.
But for separably closed fields $X(E)\subset X$ is only dense and not necessarily very dense, since there might be closed points in $X$, which are not contained in $X(E)$, so I don't know if there is a way to "gain control" over such closed points.