What's the explicit categorical relation between a linear transformation and its matrix representation?

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.

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.

Let $F$ be a field.

The category $\mathbf{Vect}_F$ of vector spaces over $F$ with linear transformations has a subcategory $\mathbf{FVect}_F$ of finite-dimentional vector spaces with linear transformation.

There's also a (small) category $\mathbf{Mat}_F$ whose objects are the natural numbers and morphisms are matrices with entries on $F$. Composition is matrix multiplication.

Now, each choice of an ordered basis for every vector space in $\mathbf{FVect}_F$ (or a full subcategory of it) gives rise to a full and faithful functor $\mathbf{FVect}_F\to\mathbf{Mat}_F$ which represents each linear transtformation by its matrix.

In particular, the fact that the functor is full and faithful means that if $V$ and $W$ are vector spaces of dimension $m$ and $n$, then $\operatorname{Hom}_{\mathbf{FVect}}(V,W)\cong \operatorname{Hom}_{\mathbf{Mat}}(m,n)$, which says that each linear transformation corresponds to exactly one matrix and vice versa.

This is, in fact, a pretty good concrete example to keep in mind of how a "full and faithful functor" behaves.

Alternatively -- especially if we have foundational quibbles over the idea of choosing a particular basis for each of the proper-class-many objects of $\mathbf{FVect}_F$ -- we could define a new cateory $\mathbf{FVectB}_F$ in which an object is a finite vector space together with an ordered basis for it. Then there is a single canonical full-and-faithful functor $\mathbf{FVectB}_F\to\mathbf{Mat}_F$. On the other hand, it feels a bit strange to introduce a distinction between the objects in $\mathbf{FVectB}$ that doesn't participate at all in deciding what the morphisms are and how they compose -- but seen at the categorical level that is really not that different from having multiple isomorphic objects in a category, which happens already for $\mathbf{FVect}$ itself.