Let $A$ be a Weil algebra. The bijection between all natural operators lifting vector fields from $m$-manifolds to the bundle functor $K^A$ of Weil contact elements and the subalgebra of fixed elements $SA$ of the Weil algebra $A$ is determined and the bijection between all natural affinors on $K^A$ and $SA$ is deduced. Furthermore, the rigidity of the functor $K^A$ is proved. Requisite results about the structure of $SA$ are obtained by a purely algebraic approach, namely the existence of nontrivial $SA$ is discussed.
We prove that the problem of finding all ${\mathcal {M} f_m}$-natural operators ${C\colon Q\rightsquigarrow QT^{r*}}$ lifting classical linear connections $\nabla $ on $m$-manifolds $M$ into classical linear connections $C_M(\nabla )$ on the $r$-th order cotangent bundle $T^{r*}M=J^r(M,\mathbb R )_0$ of $M$ can be reduced to the well known one of describing all $\mathcal {M} f_m$-natural operators $D\colon Q\rightsquigarrow \bigotimes ^pT\otimes \bigotimes ^qT^*$ sending classical linear connections $\nabla $ on $m$-manifolds $M$ into tensor fields $D_M(\nabla )$ of type $(p,q)$ on $M$.