The following is well-known.

<b>Theorem.</b> <i>Let $F\dashv U$ be a pair of adjoint functors
$$F\colon \mathcal C\to \mathcal D, \qquad U\colon \mathcal D\to\mathcal C$$
with unit $(\eta_A\colon A\to U(F(A)))_{A\in\mathcal C}$ und counit $(\varepsilon_B\colon F(U(B)) \to B)_{B\in\mathcal D}$. Then $F$ and $G$ restrict to equivalences $\mathcal C'\to\mathcal D'$ and $\mathcal D'\to\mathcal C'$ between the full subcategories $\mathcal C'$ and $\mathcal D'$ given by
$$\mathcal C':=\{A\in\mathcal C\mid \eta_A\text{ is an isomorphism}\}$$
and
$$\mathcal D':=\{B\in\mathcal D\mid \varepsilon_B\text{ is an isomorphism}\}.$$</i>

Posets can be considered as categories. This yields:

<b>Corollary.</b> <i>Let $P$ and $Q$ be posets, and $f\colon P\to Q$ and $u\colon Q\to P$ monotone maps with
$$\forall a\in P.\, \forall b\in Q.\,a\leq u(b)\iff f(a)\leq b.$$
Then $f$ and $u$ restrict to isomorphisms between the induced sub-posets $P'$ and $Q'$ given by
$$P':=\{a \in P\mid a = u(f(a))\}$$
and
$$Q':=\{b\in Q\mid f(u(b))=b\}.$$</i>

<b>Questions.</b>

1. Can someone give me the original references for the theorem and the corollary above? Of course, in some sense, the corollary must be hidden in the works of Galois (or in the works of people who cleaned up his theory), since taking $P$ to be $(\mathcal P(F), \subseteq)$ for any field $F$, $Q$ to be $(\mathcal P(G), \supseteq)$ for any group $G$ of automorphisms of $F$, and considering
$$f\colon \mathcal P(F) \to \mathcal P(G),\, a\mapsto\{g\in G\mid \forall x\in a. \, g(x)=x\}$$
and
$$u\colon \mathcal P(G) \to \mathcal P(F), \, b\mapsto \{x\in F\mid \forall g\in b. \, g(x)=x\}$$
yields the fundamental theorem of Galois theory up to a concrete characterization of $P'$ and $Q'$. But I'm interested in who extracted the "abstract content" of this proof in the sense of the corollary and who, after that, formulated the generalization to categories in the sense of the theorem.

2. Not just the relation "$g(x)=x$" between field elements $x$ in $F$ and automorphisms $g$ in $G$ yields a pair of functions $f\colon \mathcal P(F) \to \mathcal P(G)$ and $u\colon \mathcal P(G) \to \mathcal P(F)$ satisfying the conditions of the corollary, but in fact *any* relation $R\subseteq A\times B$ yields such a pair of functions $f\colon \mathcal P(A) \to \mathcal P(B)$ and $u\colon \mathcal P(B) \to \mathcal P(A)$. (Such a pair functions constituting an adjunction between posets is called a *Galois connection*, by the way.) Thus relations are a powerful tool to generate Galois connections. *Is there an analogue of relations that allows one to generate pairs of adjoint functors between categories?*

3. $P'$ and $Q'$ from the corollary can be described as the sets of *closed elements* or *fixed points* of the closure operators $u\circ f$ and $f\circ u$. Closure operators happen to be the same as monads on posets; and the algebras over a closure operator considered as a monad happen to be the closed elements of the closure operator. Incidentally, each adjunction yields a monad and a comonad and thus also algebras (coalgebras) over that monad (comonad). This makes me wonder: *Can the theorem above be formulated as an equivalence between the algebras of the monad and the coalgebras of the comonad induced by the adjunction?*

4. Not just the fundamental theorem of Galois theory can be formulated as an instance of the above theorem, but also the Gelfand–Naimark theorem, the Stone duality, the duality between affine schemes and commutative rings, and the Pontrjagin duality are corollaries of the above theorem up to concrete characterization of $\mathcal C'$ and $\mathcal D'$. I guess most of them are recasted as applications of the above theorem only with hindsight. But can the above theorem be also used as a *guide* that helps and inspires proving completely new equivalences?