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.

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.