Category theory:  There is an isomorphism between a vector space and its double-dual which does not depend on choice of basis.  It is *natural* in the sense that every vector space has such an isomorphism, and these isomorphisms commute with every linear transformation.

This should be contrasted between the isomorphisms between a finite-dimensional vector space and its dual.  These depend on a choice of basis and are not natural in this sense.

This example constitutes the first two paragraphs of the first paper in category theory!  [Eilenberg-Mac Lane: *General theory of natural equivalences*](http://www.ams.org/journals/tran/1945-058-00/S0002-9947-1945-0013131-6/S0002-9947-1945-0013131-6.pdf).

In *Categories for the working mathematician*, Mac Lane writes that the purpose of discussing categories is to discuss functors, and that the purpose of discussing functors is to discuss natural transformations.