The following are taken from "Categories" by T.S.Blyth and Arrows, Structures and Functors the categorical imperative by Arbib and Manes
$\color{Green}{Background:}$
[From: Arbib and Manes]
$\textbf{(1) Definition for “free over an object with respect to a functor":}$ 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 $G$).
[From Blyth]
$\textbf{(2) Definition:}$ Let $S$ be a non-empty $\textbf{set}$ and let $\textbf{C}$ be a concrete category. Consider the category $\textbf{K}$ whose objects are pairs $(X,\alpha)$ consisting of an object $X$ of $\textbf{C}$ and a mapping $\alpha:S\to X.$ Define $\text{Mor}_{\textbf{K}}((X,\alpha),(Y,\beta))$ to be the set of all morphisms $h:X\to Y$ in $\textbf{C}$ such that $h\circ \alpha=\beta:$
An initial object $(F,f)$ in $\textbf{K}$ (when such exists) is called a $\textit{free }\textbf{C}-\textit{object on the set S}$ .
$\color{Red}{Questions:}$
I asked here in this post about Definition (1) in regards to how to describe $GA$ for non algebraic category, like $\textbf{Top, Met}.$ It is for the purpose of determining what the free object in the category $\textbf{A}$ suppose to be. I also consulted Blyth's text and found from Definition in (2), that the pair if the pair $(F,f)$ in $\textbf{K}$ is an initial object, then it is consider to be a "free $\textbf{C}$-object" on the set $S$.
What does it mean for the pair $(F,f)$ to be consider as an initial object? Also, after determining that, will I be able to describe what $GA$ is suppose to be?
Thank you in advance.