I have a tangled web of ideas about natural transformations, vector spaces, equivalence classes, local coordinates, etc. in my head that I'm trying to unravel. So here are some of the questions I thought of:


* Vector spaces: Why is it that in linear algebra we always calculate with basis but when we think of a definition we always try to make sure it is basis independent? What kinds of symmetries are we trying to preserve by this?
* Categories: The obvious condition for natural transformations isn't so obvious to me and again I think the definition is the way it is because there is again some kind of symmetry that a natural transformation is preserving. What are the symmetries that a natural transformation is preserving?
* Equivalence classes: Almost the same thing as in the vector space scenario. We do things with representatives but we make sure our definitions are true regardless of this choice. Ok, this one is kinda silly because we are trying to preserve $\cong$ so there isn't anything too complicated happening here.
* Differential geometry: We again calculate with local coordinates but our definitions shouldn't depend on them. What symmetries are we trying to preserve in differential geometry?


So it seems to me we always break symmetry, whatever that means, when we want to work with something concrete but we try to make sure our calculations are preserved by these symmetries. Feel free to correct me and add your answers to some of my questions.


**Edit**: Dmitri gives a good answer for the definition of natural transformation in terms of the exponential and its adjoint.



Broken Symmetry
Your examples about vector spaces and differential geometry do not make any sense to me.
One does not need coordinates or bases to prove statements in linear algebra and differential geometry.
Personally, I always use coordinate- and basis-free proofs.
For me the reason to avoid coordinates and bases is that we lose geometric intuition whenever we use them.
See my manifesto on this matter here: [When to pick a basis?](https://mathoverflow.net/questions/4648/when-to-pick-a-basis/4900#4900)


One way to make the definition of natural transformation more natural is to consider the category A 
with exactly two objects and one non-trivial morphism between them.
Then the set of morphisms of an arbitrary category C is the set of functors Fun(A, C).
If we want the set of functors Fun(C, D) between two categories C and D to be a category,
then the set of morphisms of this category is Fun(A, Fun(C, D)).
But it is natural to assume that we have the standard adjunction Fun(A, Fun(C, D)) = Fun(A × C, D).
Unraveling the definition of functor from A × C to D yields precisely the usual axioms for natural transformations.


