On almost polynomial structures from classical linear connections

Let $\mathcal{M}{f}_m$ be the category of $m$-dimensional manifolds and local diffeomorphisms and let $T$ be the tangent functor on $\mathcal{M}{f}_m$. Let $\mathcal{V}$ be the category of real vector spaces and linear maps and let  ${\mathcal{V}}_m$ be the category of  $m$-dimensional real vector spaces and linear isomorphisms. Let $w$ be a polynomial in one variable with real coefficients. We describe all regular covariant functors $F:{\mathcal{V}}_m\to \mathcal{V}$ admitting $\mathcal{M}{f}_m$-natural operators $\tilde{P}$ transforming classical linear connections $\mathrm{\nabla }$ on $m$-dimensional manifolds $M$ into almost polynomial $w$-structures  $\tilde{P}(\mathrm{\nabla })$ on $F(T)M=\bigcup _{x\in M}F(T_{x}M)$.