I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way:

Equip the category $\textbf {Psh}=\operatorname{Fun}(\textbf{Ring},\textbf{Set})$ with a Grothendieck topology and define a scheme to be such an object $X$ in $\textbf {Psh}$ which has an open coverings with ring spectra (functors $\operatorname{Spec}(R)=h_R)$ and for which $h_X\colon\textbf {Psh}^{op}\to\textbf{Set}$ is a sheaf in the Grothendieck topology.

Place a Grothendieck topology on $\textbf{Ring}^{op}$ and call an object $X$ of $\textbf {Psh}$ a

*local functor*if it, viewed as a contravariant functor on $\textbf{Ring}^{op},$ is a sheaf. Then one defines open coverings in $\textbf {Psh}$ and defines shemes to be local functors wich have an open covering by spectra.

The Demazure-Gabriel volume (just about the only rather complete account of the functor-of-points approach I could find), which takes the second approach, proves what I believe is just that those two approaches are equivalent, ti. that the same objects of $\textbf {Psh}$ are schemes regardless of which the definition you follow. However, it does that by appealing to what it calls the "geometric realization" of a functor, which is just the associated ringed space. I find that very unsatisfactory and wonder if it can be proved directly in the functorial language without introducing ringed spaces?