Visualizing the correspondence between linear maps and matrices as a natural isomorphism. I was trying to interpret the equivalence between linear maps on finite dimensional vector spaces and matrices in a categorical POV. The idea is to capture the following: given a linear transformation $T: V \rightarrow W$ and fixed bases $\beta_1$ and $\beta_2$ on $V$ and $W$ respectively, there is a matrix $A \in M_{n\times n}
(K)$ s.t.
$$(T(x))_{\beta_2}
=A_{\beta_1, \beta_2} .x_{\beta_1}$$
for all $x \in V$.
It seems to me that this property can be represented by a natural transformation, but the fact that we (probably) must encode the bases in the objects or morphisms of the categories is confusing me as to how to produce categories and functors to give rise to my interpretation.
My question, summarized, is:
Is there a way to make the equation described to correspond to a natural transformation between suitable functors?
If the question is unclear, please say so and I'll try to elaborate further.
EDIT: Although I've accepted an answer which "denies" my question, investigating further I've arrived at the fact that this is indeed involves a natural transformation. The natural transformation is the sending of the ordered basis to the canonical basis of $\mathbb{K}^n$, and the functors are the identity functor and the functor described on the answer. The category is the category for which the objects are the vector spaces with a chosen ordered basis.
 A: There is a category whose objects are pairs $(V, B)$ of a finite-dimensional vector space $V$ and a basis $B$ of it, and whose morphisms $(V, B) \to (W, C)$ are linear transformations $f : V \to W$. This category is (very canonically) equivalent to the "matrix category" whose objects are the vector spaces $K^n$ and whose morphisms are linear maps $K^n \to K^m$, with the equivalence given on objects by expressing $v \in V$ in terms of the basis $B$ and given on morphisms by expressing a linear transformation $T : V \to W$ in terms of the bases $B$ and $C$. So you're looking for a functor, not a natural transformation. 
A: Although I've accepted an answer which "denies" my question, investigating further I've arrived at the fact that this is indeed involves a natural transformation. The natural transformation is the sending of the ordered basis to the canonical basis of $\mathbb{K}^n$, and the functors are the identity functor and the functor described on the answer. The category is the category for which the objects are the vector spaces with a chosen ordered basis.
