For natural numbers $r\ge 2$ and $n$ a complete classification of natural transformations $A:TT^{(r)}\rightarrow TT^{(r)}$ over $n$-manifolds is given, where $T^{(r)}$ is the linear $r$-tangent bundle functor.
@article{107719, author = {W\l odzimierz M. Mikulski}, title = {The natural transformations $TT^{(r)}\to TT^{(r)}$}, journal = {Archivum Mathematicum}, volume = {036}, year = {2000}, pages = {71-75}, zbl = {1049.58009}, mrnumber = {1751615}, language = {en}, url = {http://dml.mathdoc.fr/item/107719} }
Mikulski, Włodzimierz M. The natural transformations $TT^{(r)}\to TT^{(r)}$. Archivum Mathematicum, Tome 036 (2000) pp. 71-75. http://gdmltest.u-ga.fr/item/107719/
Natural affinors on the extended $r$-th order tangent bundles, Suppl. Rendiconti Circolo Mat. Palermo 30 (1993), 95–100. | MR 1246623
Natural operations in differential geometry, Springer-Verlag, Berlin 1993. | MR 1202431
On the order of natural operators and liftings, Ann. Polon. Math. 49 (1988), 169–178. | MR 0983220