linear algebra - 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?

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.