On the existence of some type constructions on connections

We describe all bundle functors G : FMm,n → FM such that natural bundle G2 : Mfn → FM, G2N = G(ℝm × N), G2g = G(idℝm × g) is of order 0. Next, we describe all bundle functors G : FMm,n → FM of order r such that there is a canonical construction of a general connection D(Γ, Λ) on GY → Y from a general connection Γ on p : Y → M by means of an r-th order linear connection Λ : TM → JrTM of M.