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 \(F\colon \mathcal{V}_m\to\mathcal{V}\) admitting \(\mathcal{M}f_m\)-natural operators \(\tilde{P}\) transforming classical linear connections \(\nabla\) on \(m\)-dimensional manifolds \(M\) into almost polynomial \(w\)-structures \(\tilde{P}(\nabla)\) on \(F(T)M=\bigcup_{x\in M}F(T_xM)\).