Exercise 4.2.ii from Riehl's category theory in context is as follows:
Prove that any pair of adjoint functors $F: C \rightleftharpoons D$ restrict to define an equivalence between the full subcategories spanned by those objects $c \in C$ and $d \in D$ for which the components of the unit $\eta_c$ and of the counit $\varepsilon_d$, respectively, are isomorphisms.
I'm a little confused on what it suffices to show. Denote by $C'$ and $D'$ the full subcategories defined in the question. I think one of the following should be enough but I'm not sure which:
- $F(C')=D'$ and $G(D')=C'$.
- The restriction of $F$ to $C'$ essentially surjects onto $D'$ and restriction of $G$ to $D'$ essentially surjects onto $C'$.
I've made some attempt at the first one: If $\eta_c$ is an isomorphism, then $F(\eta_c)$ is an isomorphism. I think it should be true that $F(\eta_c)=\varepsilon_{F(c)}$?
What does it suffice to show? And how can we show that?
In order to define $F'=F|_{C'}:C'\to D'$, you need to show that if $\varsigma\in C'$ then $F(\varsigma)\in D'$. Dually you need to check that $G'=G|_{D'}:D'\to C'$ makes sense.
But, given this, the natural transformations $\eta,\epsilon$ will then sensibly restrict to: $1_{C'}\implies G'F'$ and $F'G'\implies 1_{D'}$. Since they are then isomorphisms, you've already met the definition for an equivalence.
So you only have to check $F',G'$ are well-defined, really.