I am trying to understand the following definition of smooth functors on the category of finite dimensional vector spaces (in the context of smooth vector bundle theory):
Let $\mathsf{Vec}$ be the category whose objects are finite-dimensional real vector spaces, and whose morphisms are linear isomorphisms. If $\mathcal F$ is a covariant functor from $\mathsf{Vec}$ to itself, for each finite-dimensional vector space $V$ we get a map $\mathcal F:GL(V)\to GL(\mathcal F(V))$ sending each isomorphism $A:V\to V$ to the induced isomorphism $\mathcal F(A):\mathcal F(V)\to\mathcal F(V)$. We say $\mathcal F$ is a smooth functor if this map is smooth for every $V\in\mathsf{Ob(Vec)}$.
I am trying to prove the following:
Given a smooth vector bundle $E\to M$ and a smooth functor $\mathcal F:\mathsf{Vec}\to \mathsf{Vec}$, show that there is another smooth vector bundle $\mathcal F(E)\to M$ whose fiber at each point $p\in M$ is $\mathcal F(E_p)$, where $E_p$ denotes the fiber of $p$ over $M$ w.r.t. the bundle $E\to M$.
My issue is I don't quite understand how to use the smoothness of the functor $\mathcal F$ to obtain a new vector bundle $\mathcal F(E)\to M$. What are the local trivializations in such a bundle? How would one go about showing that the trivializations are diffeomorphisms?
My guess is that since we start with a smooth bundle $E\to M$, we somehow manage to express the trivializations and projections for $\mathcal F(E)\to M$ in terms of $\mathcal F$ and the ones for $E\to M$. Then smoothness of all maps would follow from smoothness of $\mathcal F$ and $E\to M$. But I have no clue how to actually accomplish this. Any help would be appreciated. TIA.