Why are natural transformations the "right" transformations between functors The concept of natural transformation is often given with no motivation for it at all. It is a strange definition that is for some reason central in category theory. But why? This question tells that natural transformations are meant to capture defining a family of morphisms $F(X)\to G(X)$ "independently of $X$". But this doesn't explain why should we consider natural transformations as the morphisms between functors.
It would seem much more natural to define morphisms of functors as commutative squares of functors, since thats the usual way to define morphisms between morphisms in categories in general. So why are natural transformations defined different? It feels like an arbitrary thing to do..
More worryingly, since natural isomorphisms are considered "the" isomorphisms between functors, it means that we use this strange definition to decide which functors to treat as "the same". In other mathematical structures like algebraic structures or models of a theory, it's clear how to define isomorphisms: these should be bijections that preserve operations and relations in both directions. Is there some similar way to define natural transformations?
 A: Since $\bf Cat$ is cartesian closed, for any two  categories $\mathbf X,\mathbf Y$ there is an exponential category  $\bf Y^X$. First notice that the objects of  $\bf Y^X$ correspond to the functors $\bf 1\to Y^X$, that by the exponential property are the functors  $\bf X\to Y$; hence if you want a "natural" notion of morphisms between such functors, you should look at what the arrows in $\bf Y^X$ are.
Similarly, the arrows of $\bf Y^X$ correspond to the  functors $\bf 2\to\bf{Y}^\bf{X}$, that are those $\mathbf{2}\times\mathbf X\to\mathbf Y $, and so those $\bf X\to Y^2$; as you see, the notion of a functor $\bf X\to Y^2$ is exactly the same as that of a natural transformation of functors  $\bf X\to Y$.
Here $\bf 1$ has one object and no non-identity arrows, while $\bf 2$ has two objects and one non-identity arrow.
A: Not really sure.
There's a little motivation from polynomial functors
$$ [S \triangleright P](x) = \Sigma s\colon S. C(P(s), x) $$
A natural transformation here is
$$ [P] \Rightarrow [Q] = \forall x. [P](x) \rightarrow [Q](x) $$
With a naturality requirement.
But this is just an unravelled version of a map between polynomials.
$$ [S \triangleright P] \Rightarrow[T\triangleright Q] = 
(f\colon S \rightarrow T)
\triangleright (Q \circ f \Rightarrow P)$$
And this directly is a structure preserving map.
There's probably a way to define functors and natural transformations nicely in terms of total functional (anafunctors) two sided discrete fibrations (profunctors).
