The complete description of all natural operators lifting real valued functions to bundle functors on fibered manifolds is given. The full collection of all natural operators lifting projectable real valued functions to bundle functors on fibered manifolds is presented.

All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^{r}E$ of E are given. Similar results are deduced for the r-jet prolongations $J_v^{r}E$ and $J^{\left[r\right]}E$ in place of $J^{r}E$.

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.

A natural $T$-function on a natural bundle $F$ is a natural operator transforming vector fields on a manifold $M$ into functions on $FM$. For any Weil algebra $A$ satisfying $dimM\ge \mathrm{w}idth\left(A\right)+1$ we determine all natural $T$-functions on $T^{*}{T}^{A}M$, the cotangent bundle to a Weil bundle $T^{A}M$.

We determine all natural transformations T²₁T*→ T*T²₁ where $T_k^{r}M=J_0^r({ℝ}^k,M)$. We also give a geometric characterization of the canonical isomorphism ψ₂ defined by Cantrijn et al.

We consider a vector bundle $E\to M$ and the principal bundle $PE$ of frames of $E$. We determine all natural transformations of the connection bundle of the first order principal prolongation of principal bundle $PE$ into itself.

We determine all natural transformations of the rth order cotangent bundle functor $T^{r*}$ into $T^{s*}$ in the following cases: r = s, r < s, r > s. We deduce that all natural transformations of $T^{r*}$ into itself form an r-parameter family linearly generated by the pth power transformations with p =1,...,r.

Let ${\overline{J}}^3$ be the functor of semi-holonomic $3$-jets and ${\overline{J}}^{3,2}$ be the functor of those semi-holonomic $3$-jets, which are holonomic in the second order. We deduce that the only natural transformations ${\overline{J}}^3\to {\overline{J}}^3$ are the identity and the contraction. Then we determine explicitely all natural transformations ${\overline{J}}^{3,2}\to {\overline{J}}^{3,2}$, which form two $5$-parameter families.

Given a map of a product of two manifolds into a third one, one can define its jets of separated orders $r$ and $s$. We study the functor $J$ of separated $(r;s)$-jets. We determine all natural transformations of $J$ into itself and we characterize the canonical exchange $J\to J^{s;r}$ from the naturality point of view.

We study geometrical properties of natural transformations $T^{A}T*\to T*T^A$ depending on a linear function defined on the Weil algebra A. We show that for many particular cases of A, all natural transformations $T^{A}T*\to T*T^A$ can be described in a uniform way by means of a simple geometrical construction.

In this paper are determined all natural transformations of the natural bundle of $(g,r)$-covelocities over $n$-manifolds into such a linear natural bundle over $n$-manifolds which is dual to the restriction of a linear bundle functor, if $n\ge q$.