Natural isomorphism are base independent isomorphisms? In Linear Algebra courses, often one hears the term "natural isomorphism" to designate isomorphism which behave like $V \cong V^{**}$ (in the finite dimensional case). Usually, one comes across the rather coarse definition: an isomorphism $\phi: V \to W$ between vector spaces is said to be natural (or canonical) when its expression does not depend on a particular choice of basis. It seems like a weird definition, to say the least, so it is tempting to look for a more rigorous one in a Category Theory book. Maclane, for example, defines firstly a natural transformation between two functors. After that, it is possible to show that if we consider the category $Vect_{K}$ of finite dimesional vector spaces, and the functors $Id, dDual: Vect_{K} \to Vect_{K}$, the first one being the identity functor, and the second one being the functor that takes vector spaces to their double dual, and linear transformations to their double dual tranformations, then we can find a natural transformation between them. 
The questions therefore are: 


*

*Is it true that a (collecion of, maybe) basis independent isomomorphism(s) "induces" a functor $\tau$ on $Vect_{K}$ in such a manner that there exists a natural transformation between $\tau$ and $Id$?

*Given a natural transformation between a functor $\tau$ on $Vect_{K}$ and the identity functor, is it true that we can find a collection of basis independent isomorphism beween vector spaces?


The second question seems stupid, but I wanted to pose the complete problem. Sorry for the long post.
 A: I think the first step to do is to make clear to yourself, that the notion of a "basis independent isomorphism" only makes sense if the isomorphism is defined "for any vector space". So in my opinion to think from the start that the basic problem is whether "two constructions lead to naturally isomorphic results", which brings you to the language of functors. It is also not necessary that one of the functors is the identity, there are lots of other nice examples. Moreover, there is no need to stick to isomorphisms, you can also think about naturally related results. 
To be concrete, consider the identity and the bidual. There is a natural transformation between these functors defined by sending a vector to the corresponding evaluation map. This is an isomorphism for finite demensional spaces but only an injection for general vector spaces, which shows that the answer to question 2 is negative. (More extremely, zero maps define - completely uninteresting - natural transformations between any kinds of functors on vector spaces.) Examples of natural transformations involving two non-trivial functors are provide by the canonical map from $V^*\otimes V$ to $L(V,V)$ which is injective in general and a linear isomorphism if $V$ is finite dimensional. There are lots of similar examples (injective in general and an isomorphism for finite dimensional spaces) related to multilinear algebra, e.g. the natural map from $S^kV^*$ to $(S^kV)^*$. 
I am not sure what to say on your question 1. I think the best way is to view a functor as a "construction" defined for general vector spaces and as a natural transformation as a "natural" way to relate two such constructions. The focus on basis independence mainly comes from the fact that using a basis is the major method to violate naturality.  
