Eilenberg and MacLane choose linear transformations to illustrate ‘natural isomorphisms’ in their famous General theory of natural equivalences. We expand here somewhat on their introductory example of the natural isomorphism between a vector space and the dual of its dual. Their example fits naturally in an axiomatic definition of vectors such as this.

Finite dimensional vector spaces are often defined as tuples of reals and then the space and its dual already seem to be identical and an isomorphism trivial and perhaps even natural. We stick to the axiomatic definition here however as it is more easily applied where needed however and we follow Eilenberg and MacLane a short distance on their famous journey.

As we have set the stag