The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes and "Theory of Mathematical Structures" by Jiří Adamek
$\color{Green}{Background:}$
We have these two definitions:
$\textbf{(1) Definition for Functors:}$ A functor $H$ from a category $\textbf{K}$ to a category $\textbf{L}$ is a function which maps $\text{Obj}\textbf{(K)}\to \text{Obj}\textbf{(L)}:A\mapsto HA,$ and which for each pair $A,B$ of objects $\textbf{K}$ maps $\textbf{K}(A,B)\to \textbf{L}(HA, HB):f\mapsto Hf,$ while satisfying the two conditions:
$H(\text{id}_A) = \text{id}_{\text{HA}}$ $\quad$ $\text{ for every }$ $A \in \text{Obj}(\textbf{K})$
$H(g\cdot f)=Hg\cdot Hf \quad\text{ whenever }g\cdot f\text{ is defined in }\textbf{K}.$
We say that $H$ is an $\textbf{isomorphism}$ if $A\mapsto HA$ and each $\textbf{K}(A,B)\to \textbf{L}(HA, HB)$ are bijections.
$\textbf{(2)Definition for "free over an object with respect to a functor":}$ 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 $G$).
We have the following three categories:
[From Arbib and Manes]
$\textbf{(3)}$ Category $\textbf{Top},$
For topological spaces, we have category $\textbf{Top},$ with topologies $\tau_X,$ $\tau_Y$ as objects and continuous maps $f:X\to Y$ as morphisms, that maps open set to open sets, $O_{\tau_Y}\in \tau_Y, O_{\tau_Y}\subset Y$ implies $f^{-1}(O_{\tau_Y})\in \tau_X, f^{-1}(O_{\tau_Y})\subset X.$
$\textbf{(4)}$ Subcategory metric space $\textbf{K}$ with lipschitz map, and $\lambda>1$:
The subcategory of Metric Space $\textbf{K}$ (contractive map) with $(X,d)$ and $(Y,e)$ defined as categories, elements of $X$ as objects, and morphisms $x\to x'$ defined as $\lambda\geq d(x,x')$
[From Adamek]
$\textbf{(5)}$ Category metric space $\textbf{Met}$ with lipschitz map $\lambda=1$:
Category of $\textbf{Met}:$ The construct $\textbf{Met}$ has as objects all $\textit{metric spaces,}$ i.e., pairs $(X,\alpha)$ where $X$ is a set and $\alpha$ is its metric. The morphisms from a space $(X,\alpha)$ to a space $(Y,\beta),$ called $\textit{contractions,}$ are maps
$f:X\to Y$
such that
$\beta(f(x_1),f(x_2))\leq \alpha(x_1,x_2)\quad \forall x_1,x_2\in X.$
$\color{Red}{Questions:}$
I am having difficulty figuring out what the codomain $GA$ of the morphism $\eta \in \textbf{B}$ is suppose to be in definition $\textbf{(2)},$ for the three non algebraic categories listed in $\textbf{(3)},\textbf{(4)},$ and $\textbf{(5)}.$ For all three cases, we can let $\textbf{B}=\textbf{Set},$ the category of set, and $B$ to be an object in $\textbf{B}.$ Both $A, A'$ are objects in $\textbf{A}.$ In the context of the three categories $\textbf{(3), (4),(5)},$ category $\textbf{A},$ depending on which of those three is to be respectively $\textbf{Top, K, Met.}$ with objects $A,A'$ in $\textbf{A}.$ In the context of those three categories the objects $A,A'$ are also respectively: $\tau_X, \tau_Y, x_1,x_2, (X,\alpha), (Y,\beta).$ We have the morphism $\psi\in \textbf{A},$ and the functor $G\psi:GA\to GA'.$ I don't know how to describe $GA, GA',$ for each of the three categories, even though I know how to write the functor for each of them. Hence I don't know what $\eta(b), b\in B$ suppose to equal to other than that $\eta(b)\in GA,$ and for the same reason, $f(b)\in GA'.$ Thank you in advance.