The following are taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes.
$\color{Green}{Background:}$
$\textbf{Definition for “free over an object with respect to a functor":}$ Let $G:\textbf{A}\to \textbf{B}$ be any functor, and $B$ an object of $\textbf{B}.$ We say the pair $(A,\eta),$ where $A$ is an object of $\textbf{A}$ and $\eta:B\to GA$ is a morphism of $\textbf{B},$ is $\textbf{free over}$ $B$ $\textbf{with respect to}$ $G$ just in case $\eta;B\to GA$ has the couniversal property that given any morphism $f:B\to GA'$ with $A'$ any object of $\textbf{A,}$ there exists a unique $\textbf{A-}$morphism $\psi:A\to A';$ such that
Diagram 1
We refer to $\eta$ as the $\textbf{inclusion of generators;}$ and call the unique $\psi$ satisfying the above diagram the $\textbf{A-morphic extension of } f$ (with respect to $G$).
$\color{Red}{Questions:}$
If we let $\textbf{A}=\textbf{Top}$ and $\textbf{B}=\textbf{Set}$ in Definition above, and $G$ to be the forgetful functor. We let $(A‚\tau_A),(A',\tau_{A'})$ be topological spaces which are objects standing for $A,A'$ respectively in $\textbf{Top}.$ Let $\psi:(A‚\tau_A)\to(A',\tau_{A'}),$ be a morphism such that $\psi\in \textbf{Top}((A‚\tau_A),(A',\tau_{A'}))$ and $G\psi\in \textbf{Set}(G(A‚\tau_A),G(A',\tau_{A'}))$ be the functor $G$ in $\textbf{Top}((A‚\tau_A),(A',\tau_{A'}))\xrightarrow{G\psi} \textbf{Set}(G(A‚\tau_A),G(A',\tau_{A'})).$ As a functor, it is defined on objects in $\textbf{Top},$ $(A‚\tau_A)\mapsto G(A‚\tau_A), (A',\tau_{A'})\mapsto G(A',\tau_{A'}).$ But because $G$ forgets the structure of the objects in $\textbf{Top},$ then $G(A‚\tau_A)=A, G(A'‚\tau_{A'})=A'.$ Also for the case of what $G$ does to morphisms, $G\psi=\psi.$ Now let $B$ be an object in $\textbf{Set},$ and $GA=G(A‚\tau_A)=A,\eta:B\to GA:=\eta:B\to G(A‚\tau_A):=\eta:B\to A.$ Similarly, $f:B\to GA':=f:B\to A'.$ I would like to know if my understanding of the $\eta$ map is correct given that $G$ is a forgetful functor.
Thank you in advance.