I am working on the following exercise in Ravi Vakil's algebraic geometry notes ("The Rising Sea")
Show that the map $\tau_{AB}$ (1.5.0.1) has the following properties. For each $A$ there is a map, $\eta_A: A \to GF(A)$ so that for any $g: F(A) \to B$ the corresponding $\tau_{AB}(g):A \to G(B)$ is given by the composition $$A \overset{\eta_a}{\longrightarrow} GF(A) \overset{Gg}{\longrightarrow} G(B)$$
$\textbf{My Attempt:}$
Since $$\tau_{A,F(A)}: Mor(F(A), F(A)) \to Mor(A, GF(A))$$
Thus for any $\eta_A$, there is a corresponding $k \in Mor(F(A), F(A))$ since $\tau_{A,F(A)}$ is a bijective correspondence.
Now, $Id_{F(A)} \in Mor(F(A), F(A))$.
This implies (?) that there must exist $k \in Mor(A, GF(A))$ such that $k=\tau_{A,F(A)}(Id_{A})$.
If this is true, can we say $\eta_A = \tau_{A,F(A)}(Id_{A})$ ?
This is currently exercise 1.5.B on August 2022's draft of Ravi Vakil's FOA. First, I am going to state it properly.
We're given a pair of adjoint covariant functors $(F,G)$, whose natural bijection is given by the maps $\tau_{AB}:Mor_B(F(A),B)\rightarrow Mor_A(A,G(B))$. We have to prove that, for each $A$ in $\mathcal{A}$, there exists a map $\eta_A:A\rightarrow GF(A)$ such that, for any map $g:F(A)\rightarrow B$ in $\mathcal{B}$, the corresponding $\tau_{AB}(g):A\rightarrow G(B)$ is given by the composition \begin{equation*} A\overset{\eta_A}{\longrightarrow}GF(A)\overset{Gg}{\longrightarrow}G(B). \end{equation*}
As you did, and since we are looking for a map $\eta_A:A\rightarrow GF(A)$, it makes sense to consider the map $\tau_{AF(A)}:Mor_B(F(A),F(A))\rightarrow Mor_A(A,GF(A))$.
The key step is to realize that we want to prove that a map $\tau_{AB}(g)$ can be expressed as a composition, and that we can do that for any $g:F(A)\rightarrow B$ in $\mathcal{B}$. This makes us think about the naturality conditions for arrows on $\mathcal{B}$ with the maps $\tau_{AF(A)}$. These, for any $g:F(A)\rightarrow B$ give us commutative squares
$\require{AMScd}$ \begin{CD} Mor_B(F(A),F(A)) @>g>> Mor_B(F(A),B)\\ @V \tau_{AF(A)} V V @VV \tau_{AB} V\\ Mor_A(A,GF(A)) @>>Gg> Mor_A(A,G(B)). \end{CD}
If we plug in on the upper left corner the identity $Id_{F(A)}$ we obtain the the desired $\eta_A:=\tau_{AF(A)}(Id_{F(A)})$ and the condition that was asked for (note that $\eta_A$ does not depend on $g$).