Yoneda lemma applied to finite dimensional vector spaces. This question is stenmed from my attempts to review my underestimating of the basic theory of finite dimensional vector spaces in a categorical language, and particularly to highlight he functorial behavior of the concept of base. My hope is to see  possible corrections, clarifications or link to a more through expansions of such attempts:
Background:
A vector space is finite dimensional if it has a finite spanning subset. Let us denote the category of finite dimensional vector spaces and linear transformations by $\mathcal{V}$ and the well known category of matrices by $\mathcal{M}$, assuming that the underlying field is understood. Recall that the object of $\mathcal{M}$ are natural numbers and morphisms from $m$ to $n$ are matrices with $m$ rows and $n$ columns. (Fortunately, the matrix multiplication and the famous identity $I^{n\times n}$ matrices composition satisfy the axioms of category). 
Question:

Obviously, a base has a functorial nature but how one should introduce the concept of base, purely in categorical language? Is the base a representable functor and does the canonical way of obtaining a (standard) base has anything to do with Yonada lemma?

Thanks.

There is a distinguished $1$-dimensional vector space called $k$, the base field. This $1$-dimensional vector space is distinguished by the fact that it is equipped with canonical isomorphisms
$$k \otimes V \cong V \cong V \otimes k$$
satisfying certain identities making $k$ the monoidal unit for the tensor product. These canonical isomorphisms, among other things, distinguish the basis vector $1 \in k$. The reason is that the above isomorphisms specify actions of $k$ on each finite-dimensional vector space $V$ and $1 \in k$ is uniquely specified as the element of $k$ which acts by the identity. 
A basis of a finite-dimensional vector space $V$ is a choice of isomorphism
$$k \oplus ... \oplus k \to V.$$
Here it is crucial that we use $k$ (equipped with the maps above) and not just a $1$-dimensional vector space. Beyond this it is not clear to me what you mean by "a base has a functorial nature." 
