On the notion of "naturally in" (within the context of natural isomorphisms)

Question

I currently work through Tom Leinster's Basic Category Theory and I am not sure if I really grasp the notion of naturally in $A$. Since this sentence does not make any sense stated isolated I will add some context.

Defintion $1.3.12$ Given functors $\mathscr A \underset{\small G}{\overset{\small F}{\large{\rightrightarrows}}}\mathscr B$, we say that $$F(A)\cong G(A)~\textbf{naturally in}~A$$ if $F$ and $G$ are naturally isomorphic$^1$.

Up to this point I do not really understand why it is necessary to introduce this new piece of terminology. From what I can tell we are perfectly fine with the notion of natural isomorphism yet alone. Leinster adds an explanation why we actually need this extended terminology.

This alternative terminology can be understood as follows. If $F(A) \cong G(A)$ naturally in $A$ then certainly $F(A) \cong G(A)$ for each individual $A$, but more is true: we can choose isomorphisms $\alpha_A : F(A) \to G(A)$ in such a way that the naturality axiom $(1.3)^{2}$ is satisfied.

I have got the feeling that I do not really understand this paragraph. Apparently, it is not enough for two functors to be naturally isomorphic yet alone to guarentee that there exists a isomorphism for each individual object $A$. But I am not sure why this is so.
My wild guess is that, in analogy to what is later on explained (we need a functor to be not only faithful and full but also essentially surjective to achieve an equivalence between categories, since we need to get a transformation for all objects) that Definition $1.3.12$ enables us to guarentee the existence of a natural isomorphism between $F$ and $G$ that holds for all objects of $\scr A$ and not only for a smaller portion of the (e.g. a full subcategory of $\scr A$).

Is my intuition right? If not, could someone please illuminate me about this (strange) definition?

Thanks in advance!

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$^1$ Wihin the book a natural isomorphism between functors from $\scr A$ to $\scr B$ is an isomorphism in $[\scr A,B]$ (the functor category). [Definition $1.3.10$]

$^2$ The naturality axiom states that for every map $A\overset{f}\to A'$ the square $$\require{AMScd} \begin{CD} F(A) @>{F(f)}>> F(A')\\ @V{\alpha_A}VV @VV{\alpha_{A'}}V \\ G(A) @>>{G(f)}> G(A') \end{CD}$$

commutes, where $\alpha:F\to G$ is a natural transformation (and $F,G:\scr A\to B$).

Answer 1

"$F(A)\cong G(A)$ naturally in $A$" means exactly the same thing as "$F$ and $G$ are naturally isomorphic".

The annotation "natural in $(-)$" starts pulling its weight for multivariable functors of mixed variance. Then, once we have a notion of "contravariant naturality", we can write succinct statements like

$\mathrm{ev}:B^A\times A\to B$ is natural in $A$ and $B$

or

$\circ:B^A\times C^B\to C^A$ is natural in $A$, $B$, and $C$

Answer 2

I'm trying to understand the same question in the statement and proof of the yoneda lemma in the same book. My take as a programmer (though I did maths in Uni) is that initially the identifiers (in the yoneda case: A and X) start out as fixed things - i.e. they are parameters in that part. Then when we get to the "naturally in" bit they switch seemlessly to being free variables that are just there as placeholders to show what partial functions we are actually talking about.