Lawvere: Recovering Galois connections from "globalized" Galois connections (adjunctions) In the section on globalized Galois connections in Lawvere's paper Adjointness in foundations, Lawvere suggests to replace Galois connections (which are, adjunctions between poset-categories) by more profound adjunctions between non-poset-categories containing more information, of which the former are "fragments".
As an example, he mentions that the familiar Galois connection between intermediate fields of $L/K$ and subgroups of $Gal(L/K)$, for some fixed Galois extension $L/K$, can be "globalized" to an adjunction between the opposite category of the category of $k$-algebras and the category of topological $Gal(\overline{k}/k)$-sets.
Question: In which sense can the Galois connection be considered as a "fragment" of the adjunction? Lawvere writes

Restricting to subalgebras of $\overline{k}$ on the one hand
and to quotients of the regular representation on the other retrieves in effect the usual
Galois connection.

but I don't understand this and how this "retrieves in effect the usual
Galois connection".
Another example he mentions is that the Galois connection between ideals in $k[x_1, \dots, x_n]$ and varieties can be "globalized" to the adjunction between $k$-algebras and schemes (given by "spectrum" and "global section"). Here I also wonder: how can the former be recovered from the latter?
 A: In Lawvere's globalized Galois correspondence, the $K$-algebra $\overline{K}$ corresponds to the regular representation of the Galois group, i.e. the action of $G=\operatorname{Gal}(\overline{K}/K)$ on itself.
The point is this: our adjunction induces an adjunction between the category of monomorphisms $E\to \overline{K}$ of $K$-algebras and the category of epimorphisms $G\to S$ of (topological) $G$-sets.  We use here the fact that adjoint functors play nicely with monos and epis. (It is an exercise for the reader to check that these induced functors still form an adjoint pair.)
But any $K$-subalgebra of $\overline{K}$ is a field, and any quotient of $G$ is determined up to isomorphism by its stabilizer $H$, which can be any (closed) subgroup of $G$.
So, replacing our categories by equivalent ones, we now have an adjunction between the category of intermediate fields $\overline{K}/E/K$, and the category of (closed) subgroups $H\leq G$.  Both are just posets, so this is a traditional Galois connection.
(N.B. To avoid the categorical machinery we could just check directly that our functors behave correctly on the subfields of $K$ and subgroups of $G$, but I feel that it's good to make clear that we can get here with only general properties of adjoint functors.)
To get the version you mention, all we do is restrict these categories further, to consider only subfields of some Galois $L/K$, and the category of subgroups of $\operatorname{Gal}(L/K)$.  Actually, we can repeat the above argument with $\overline{K}$ replaced by $L$, and in the finite case one can even omit the topological bits.
As for your example from algebraic geometry, I think the basic idea is that a closed embedding of schemes $f:X\to Y$ can be described by a surjection of sheaves $\mathcal{O}_Y\to f_\ast \mathcal{O}_X$, so you can think of closed subschemes of $Y$ as sheaves of ideals of $\mathcal{O}_Y$.  Since a variety is just a closed subscheme of $\mathbb{A}^n$ (more or less), the global correspondence should tell us how to relate these to quotients, thus ideals, of $k[X_1,\ldots , X_n]$.
