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.