Let $F: A \to B$ and $G: B \to A$ be adjoint functors, with $F \dashv G$. There is a full subcategory $A'$ of $A$ consisting of those objects $a$ for which the unit map $a \to GF(a)$ is an isomorphism, and there is a dually-defined full subcategory $B'$ of $B$. It is an elementary exercise to show that $F$ and $G$ restrict to an equivalence $A' \simeq B'$.

Either of the equivalent categories $A'$ and $B'$ is called the invariant part or fixed category of the adjunction. There are other names too; the terminology hasn't settled down.

Q. Where did this general construction first appear in print?

Adjoint functors were introduced by Kan in 1958. I don't see this construction in his paper. But I guess someone must have mentioned or used it quite soon thereafter. I want to know who I should cite.

(Let me make clear that I'm not asking about particular instances of this construction. It's the general construction I'm after.)