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 $\eta_A = \tau_{A,F(A)}(Id_{A})$ ?