'No arbitrary choices' intuition for natural transformation.

Question

One of the most frequent examples for natural transformation is the natural isomorphism between vector spaces and their double duals given by evaluation map $$v \to eval_v$$ and it is given in contrast of isomorphism between $V$ and $V^{*}$ where for construction of isomorphism we should choose some basis. But aren't we making arbitrary choice in natural case too when we opt to do $$v \to eval_v$$ instead of, for example $$v \to 2\ eval_v$$ ? Can someone elaborate, what am I missing here and why one choice is more arbitrary than the other?

Answer 1

A mathematically precise notion of natural transformation in full generality is given by category theory. I won’t get into the details here but you can find the details in any textbook (e.g., Category Theory in Context, available online). The gist is that the morphisms, in your case linear transformations, are part and parcel of a construction being natural or not. In this case, the natural isomorphism between a finite dimensional vector space and its double dual is such that that particular choice is coherently compatible with all of the linear transformations in existence. Intuitively, if a construction depends on arbitrary choices, then those choices will sabotage this compatibility for at least one linear transformation. Again, details in any textbook.

Answer 2

You could map to twice the evaluation morphism, and that would give you a different natural transformation. But if you do that, you need to do it everywhere, or the connecting morphisms wouldn't match up in the required way.

So the intuition is actually that there's no room for making a new independent arbitrary choice for each object.