What is a natural isomorphism?
I have often encountered the phrase "There is a natural [way/map/etc.]..." when describing say isomorphisms, maps, etc. What exactly does "natural" mean?
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As the above comments allude to, mathematicians often use the word informally. It's often used in the sense that the way/map/whatever the author is describing is the one that most people reader the paper would expect it to be (though this can be a little frustrating if one disagrees). It can also mean that the way/map/whatver is independent of any choices that are implicit in its definition. An example would be the isomorphism between a vector space and its double dual, which doesn't require specifying a basis.
There is also a technical usage of the term, coming from category theory. See http://en.wikipedia.org/wiki/Natural_transformation. This definition is essentially a way of making the 'independence-from-choices' notion rigorous.