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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?

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    $\begingroup$ It's certainly not true in any reasonable sense that a natural transformation is one that "makes no arbitrary choices". This is at best a very loose intuition. $\endgroup$ – Eric Wofsey Jun 16 at 21:28
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    $\begingroup$ There is actually no natural isomorphism between vector spaces and their double duals. There is a natural morphism, but it is an isomorphism only for finite dimensional spaces. $\endgroup$ – Marc van Leeuwen Jun 17 at 8:46
  • $\begingroup$ Are you perhaps thinking of the term "canonical" which is an imprecise and colloquial way of saying something akin to "we didn't need any choices, this thing exists of necessity by prior assumptions alone"? For vector spaces we often have a choice of basis, and that's how we normally construct an isomorphism $V\to V^*$ for $V$ finite dimensional. But a basis is "extra" information we have to introduce on top of the vector space; while the double dual isomorphism does not make use of a basis, just the vector space, and is in some sense "obvious", so is often referred to as "canonical". $\endgroup$ – zibadawa timmy Jun 17 at 9:32
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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.

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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.

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