TITLE: Are all vector-space valued functors on sets free?
QUESTION [14 upvotes]: Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between them. There is a free functor $G \colon \mathbf{Set} \to \mathbf{Vect}$, left adjoint to the forgetful functor, given by $G(X) = \mathbb{C}^X$ and $G(f)(\phi)(y) = \sum_{f(x)=y} \phi(x)$.

Is any functor $\mathbf{Set} \to \mathbf{Vect}$ of the form $H \circ G \circ F$ for $F \colon \mathbf{Set} \to \mathbf{Set}$ and $H \colon \mathbf{Vect} \to \mathbf{Vect}$?

A priori one might expect lots more functors, but I'm having a hard time coming up with any. On the other hand, functoriality seems to keep $G$ from "creating chaos" to "mess up freeness" (sorry that I can't explicate this feeling better). To keep it simple, let's keep things finite(-dimensional), and not consider anything about other base fields, or monoidal structure.

REPLY [19 votes]: The easiest example is the functor that sends the empty set to a 1-dimensional vector space, and all other finite sets to zero.  (Any H that sends some vector space to 0 must send 0 to 0 as well).
The second-easiest example is the one given by Jeremy Rickard.
If you're interested in functors from the category of finite sets to finite dimensional vector spaces, you might take a look at my paper http://arxiv.org/abs/1406.0786
The main result is a structure theorem about (finitely-generated) functors in this category.