I'm learning algebraic geometry and I'm trying to reconcile the locally ringed space and functor of points perspectives. Often when one defines a morphism $X \to Y$ of schemes thought of as locally ringed spaces they begin by saying where $x \in X$ goes to $y \in Y$, and then show that the resulting map is actually continuous, and then a morphism of locally ringed spaces.
(From now on, let's just suppose that $X = \operatorname{Spec} A$ is an affine scheme.)
On the other hand if $X$ and $Y$ are schemes thought of as functors of points, a morphism $X \to Y$ is just a natural transformation of functors from rings to sets. I figure that a recipe to take the point $x$ of a locally ringed space $X$ to a point $y \in Y$ of another translates into the functor of points perspective as a rule which produces from any field point $\operatorname{Spec} K \to X$ a field point $\operatorname{Spec} K \to Y$.
Of course, this is on its face not a rule to perform this translation for an arbitrary $R$-points $\operatorname{Spec} R \to X$. Nonetheless, by the equivalence between the two perspectives we should have:
- that any natural transformation $X \to Y$ of functors of points is completely determined by what happens to the field points plus some additional gluing data (encoded in the LRS language as a morphism of structure sheaves), and
- the conditions "is continuous", and "is a morphism of locally ringed spaces", should translate into conditions on the map of field points and additional gluing data required for the rule to assemble into an entire natural transformation $X \to Y$.
Is someone able to explain how this correspondence goes for (1) and (2) (or direct me to a reference)? Is there a convenient way to formulate how the additional gluing data (besides what happens to field points) is expressed?