TITLE: What's the explicit categorical relation between a linear transformation and its matrix representation?
QUESTION [6 upvotes]: There several questions about linear transformations and its respective matrices in some basis, but I'm particularly interested in the explicit definition of this relation in the category $Vect$ (of vector spaces and linear transformations.) Of course, a inevitable question is if there is always an matrix (or tensor, perhaps) representation (representable Functor?) for every morphism in an arbitrary category.
Sorry if it is a silly question, but I'm learning CT by myself and it is difficult for me to understand this relation in categorical terms.

REPLY [10 votes]: Let $\Bbbk$ be a field and $\mathsf{Vec}_f$ the category of finite dimensional vector spaces over $\Bbbk$. Let also $\mathsf{Mat}$ be the category whose objects are natural number $n \ge 0$, morphism $n \to m$ are matrices $m \times n$ with entries in $\Bbbk$, and composition is matrix multiplication.
There is a functor $F : \mathsf{Mat} \to \mathsf{Vec}_f$ given by $F(n) = \Bbbk^n$ and $F(M)$ is the linear map represented by $M$ in the standard bases of $\Bbbk^m$ and $\Bbbk^n$.
Then this functor is full and faithful: the induced map $F : \hom_\mathsf{Mat}(m,n) \to \hom_{\mathsf{Vec}_f}(\Bbbk^m, \Bbbk^n)$ is a bijection (this is a standard fact about linear algebra). It's also essentially surjective: every finite dimensional vector space is isomorphic to $\Bbbk^n$ for some $n$. Therefore it's an equivalence of categories.

This is very similar to Henning Makholm's answer, but his is mixing up two issues: the existence of the equivalence of categories $F$, and the existence of a pseudo-inverse to an equivalence of categories using some kind of choice.