how are these two different definitions for free objects over an object the same?

Question

The following are from: "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes, "Theory of Mathematical Structures" by Jiří Adamek

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[From Adamek]

$\textbf{(1a) Definition:}$ An object $(X,\alpha)$ is said to be $\textit{free over a set}$ $M\subset X$ provided that for each object $(Y,\beta)$ and each map $f_0:M\to Y$ there exists a unique morphism $f:(X,\alpha)\to (Y,\beta)$ extending $f_0$ (i.e., with $f(m)=f_0(m)$ for all $m\in M)$

$\textbf{(1b) Definition:}$ A construct is said to have $\textit{free object}$ if for each cardinal number $n$ there exists a free object on $n$ generators.

[From Arbib and Manes]

$\textbf{(2) Definition:}$ 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 $G$).

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I would like to know how the definition from (1a) and (1b) differ from (2). Both definition uses morphisms but the second is worded using the idea of functors. If they are the same, can someone also please explain how the two are equivalent.

Thank you in advance

Answer 1

In the book Category Theory of Awodey, section 1.7 (Free categories), he points out how this definition works in the specific case of monoids. For me it's the best place to learn this, it's very didactic. Also he generalize it in the chapter he speaks about adjunction.

But to spit out how definition 1 is a special case of 2, as commented, we can rename the definition 2 with the names of definition 1.

In definition 2, let $G: A \to (B = Set)$ be the forgetful functor (we can use it because in definition 1 there is a underlying set in each object of the algebraic structure), that sends each $A$-object, $(X,\alpha)$, to the set $X$, also it sends each $A$-morphism $f:(X, \alpha) \to (Y, \beta)$ to the $Set$-morphism $f_0: X \to Y$ (careful here, $f$ is the morphism of the structured set of the definition 1, not the one of the definition 2, that we will write in bold f).

Translating the terms of definition 2 to the definition 1:

1. the object $A$ will be $(X, \alpha)$,

2. $A' = (Y, \beta)$

3. $GA = X$

4. $GA' = Y$

5. $B = M$

6. $\psi = f$

7. $G\psi = Gf $

8. $\eta $ (not mentioned)

9. f $= f_0$

Now read the definition 2 with this translations. Let say $\eta = i$ (of insertion of the generators of $M$ in $X$), such that $i:M \to X$ is a inclusion. Then, you can see that the diagram of definition 2 is commutative iff in the point $m$:

$$f_0(m) = Gf \circ i(m) = Gf(m) = f(m)$$

that is the condition Adamek say it is the extension property.