I am looking for a reference where the notion of "fixed point of an adjunction" is examined extensively; by this I mean that given an adjunction $F\colon {\cal C\leftrightarrows D}\colon G$ I'm looking for a description of the subcategories of those $c\in \cal C$ such that $c\cong GFc$, and $d\in\cal D$ such that $FGd\cong d$.
I recall a paper on TAC beginning precisely with this definition, maybe the origin of this quote by M. Brandenburg?
Is there something more than the equivalence between $Fix(FG)$ and $Fix(GF)$ that can be said in general? Google is really poor of results.
I arrived to this notion wondering how to characterize those (c,d) in CxD such that the adjunction bijection $$ \hom(Fc,d)\cong \hom(c, Gd) $$ restricts to a bijection $Iso(Fc,d)\cong Iso(c, Gd)$, i.e. those $(c,d)$ such that $Fc\to d$ is an iso iff its mate $c\to Gd$ is an iso. This entails easily that $c\in Fix(GF)$ and $d\in Fix(FG)$.
