Maps between general linear group that can be extended to functor this is just a basic linear algebra question, which I do not have a idea.
Suppose that we have a group homomorphism $\phi:GL_n(\mathbb{R})\rightarrow GL_m(\mathbb{R})$. Is there always a functor $F:Vect_n \rightarrow  Vect_m$ where for any $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$, $F(f)=\phi(f)$? 
Clearly if there is a canonical choice of a basis for each vector space then we can define a functor. But I am thinking of defining a functor in an intrinsic way, like $V^{\otimes m}$, $\textrm{Sym}^nV$, $\Lambda^n V$ etc.. Is it always possible to construct $F$ is this way?
Thank you.
 A: If we assume $Vect_n$ means the category whose objects are $n$-dimensional real vector spaces and whose morphisms are $\mathbb{R}$-linear maps, then your question doesn't quite make sense, because the notation $\phi(f)$ requires $f$ to be invertible.  If instead we assume $Vect_n$ means the core of the previous category, i.e., morphisms are isomorphisms of real vector spaces, then a positive answer to your question follows from global choice.  This is essentially your reasoning involving a basis: we may define a functor on a skeleton of the source category and choose a noncanonical extension.
The usual method for expressing naturality of a representation is to describe it as the restriction of an endofunctor on the category of all finite dimensional real vector spaces, instead of restricting to a single dimension.  The representations you get are precisely those defined by polynomial equations (i.e., algebraic representations), and the functors are direct sums of Schur functors.  (Edit: A previous version of this paragraph restricted to additive endofunctors, but these are uniquely determined by what happens to the one dimensional vector space, and hence a bit too restrictive.)
In general, you can get many other representations.  For example, we may choose a discontinuous automorphism $\psi$ of $GL_1(\mathbb{R})$, and take the tensor product of any nice map $GL_n \to GL_m$ with the composite of determinant with $\psi$, to get a rather awful object.  Even with continuous homomorphisms, one can get things like $A \mapsto \begin{pmatrix} 1 & s \log |\det A | \\ 0 & 1 \end{pmatrix}$ for real numbers $s$, so I doubt there is a good classification.
