Let $k$ be a field and $F_n$ be a finitely generated $k$-algebra with $n$ generators. Then, a $F_n$-module is the same thing (more precisely, there is an isomorphism of categories) as a $k$-vector space $V$ with $n$ endomorphisms $(\varphi_j)_j$.
Adopting the second viewpoint, a module homomorphism $f:V \to W$ is a $k$-linear map with $f \phi_j = \phi_j f$.
If we draw out the diagram we would realize that this is exactly the diagram for a natural transformation. We somehow would want a functor to just hit a single object (namely $V$ or $W$).
Is there any natural way to make this precise? I can construct functors such that this viewpoint works but those functors are not really natural and don't seem to be of any relevance at all. So is there any way to interpret the structure of a module (over an algebra) categorically?
There’s no need for finite generation hypotheses or anything like that. An arbitrary $k$-algebra can be viewed as a $k$-linear category with one object, and then a functor into vector spaces is identified with a module, while a natural transformation is identified with a module map. This is nothing deep-it’s essentially just rewording the description of an $A$-module as a vector space equipped with a homomorphism from $A$ to its endomorphism algebra. On the other hand it has the nice consequence of easily generalizing to let you talk about modules over “$k$-algebras with several objects”, i.e. arbitrary small $k$-linear categories.