When we say some arrow $\eta _A$ is natural in $A$ ($A$ being an object of the category in question, $\mathsf C$), we mean it is a component of a natural transformation. I have consistently stumbled upon statements in the spirit of:
The natuality of $\eta _A$ in $A$ means that the object $A$ is dispensable and is merely a representative of objects in $\mathsf C$.
I'm trying to understand exactly how this follows from the definition. The nicest formulation for the definition I have come up with is the following:
For each arrow class $\mathsf {Hom}_{\mathsf C}(A,B)$ there are two arrows $\eta _A:FA\rightarrow GA$ and $\eta _B:FB\rightarrow GB$ that make the usual square commute independently of which arrow we take in $\mathsf {Hom}_{\mathsf C}(A,B)$.
The only systematic aspect I see here is the independence of $\eta _A$ and $\eta _B$ of the arrow $f\in \mathsf {Hom}_{\mathsf C}(A,B)$ whose images we take by $F$ and $G$ to make the commutative square. I do not, however, see how this definition ignores particular properties of the objects $A,B$.
So where does this notion of naturality hide in the definition of natural transformations?
Big Addition: As my initial question was probably too vague, I'm adding a excerpt from Marquis's From a Geometrical Point of View - A Study of the History and Philosophy of Category Theory. I do not understand how some portions (underlined in blue) follow from the formal definitions, even heuristically. I would very much like explanations because according to the underlined bits, naturality is exactly what we'd like it to be.
The words "dispensible" and "merely a representative" are just informal descriptions of the definition of a natural transformation. Therefore you won't be able to find a "proof" from the definition. The definition says what it says and it cannot really be simplified. Another informal description is that the $\eta_A$ vary "continuously with $A$". This is similar to the notion of a homotopy (cf. MO/64365).