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.