This question is about a problem from here. I'll restate it:
Let a pair of functors $F : \mathscr{A} \rightarrow \mathscr{B}$ and $G : \mathscr{B} \rightarrow \mathscr{A}$ be adjunction such that $F$ is left adjoint to $G$, i.e. $F \dashv G $. Write $\textbf{Fix}(GF)$ for the full subcategory of $\mathscr{A}$ whose objects are those $A \in \mathscr{A}$ such that $\eta_{A}$ is an isomorphism, and dually $\textbf{Fix}(FG) \subseteq \mathscr{B}$. Prove that the adjunction $(F, G, \eta, \epsilon)$ restricts to an equivalence $(F', G', \eta', \epsilon')$ between $\textbf{Fix}(GF)$ and $\textbf{Fix}(FG)$.
I posted an answer in the link above, but it's incomplete. Namely, it doesn't prove that $F'(A)\in \mathbf{Fix}(FG)$, that $F'(a\to a')\in \mathbf{Fix}(FG)(F'(a),F'(a'))$ for any arrow $(a\to a')\in \mathbf{Fix}(GF)(a,a')$, and similarly for $G'$.
Here is my attempt to prove the above facts.
(1) Let $A\in \mathbf {Fix}(GF)$. We show that $F'(A)\in \textbf{Fix}(FG)$. The assumption means that $\eta_A:A\to GF(A)$ is an isomorphism. And we need to prove that $\epsilon_{F(A)}: FGF(A)\to F(A)$ is an isomorphism. Since $F\dashv G$, the following diagrams commute (Leinster, Lemma 2.2.2):
Look at the left diagram. Since $\eta_A$ is an isomorphism, then so is $F(\eta_A)$ since functors preserve isomorphisms. The arrow $1_{F(A)}$ is an isomorphism too. So the vertical arrow $\epsilon_{F(A)}$ must be an isomorphism too, q.e.d.
Is this argument correct?
(2) Suppose $a\to a'$ is an arrow in $\mathbf{Fix}(GF)$. We need to show that its image under $F'$ is in $\mathbf{Fix}(FG)$. But since $\mathbf{Fix}(GF)$ is full, the set of arrows in $\mathscr A$ is equal to the set of arrows in $\mathbf{Fix}(GF)$. So $F'(a\to a')=F(a\to a')\in \mathscr B(F(a),F(a'))$. But $\mathscr B(F(a),F(a'))$ is equal to $\mathbf{Fix}(FG)(F(a),F(a'))$ since $\mathbf{Fix}(FG)$ is full.
Is this argument correct?
(3) Finally, we need to prove the analogous statements for $G'$. I can do a proof similar to (1), but can I somehow appeal to duality without doing the proof again? How exactly?