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If m≥3 and r≥0, we deduce that any natural linear operator lifting vector fields from an m-manifold M to the r-jet prolongation JʳTM of the tangent bundle TM is the composition of the flow lifting Jʳ corresponding to the r-jet prolongation functor Jʳ with a natural linear operator lifting vector fields from M toTM. If 0≤s≤ and m≥3, we find all natural linear operators transforming vector fields on M into base-preserving fibred maps JʳTM→JˢTM→JˢTM

Lifting vector fields from manifolds to the $r$-jet prolongation of the tangent bundle