Let $r,s,q,m,n\in \mathbb{N}$ be such that $s\ge r\le q$. Let $Y$ be a fibered manifold with $m$-dimensional basis and $n$-dimensional fibers. All natural affinors on ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$ are classified. It is deduced that there is no natural generalized connection on ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$. Similar problems with ${\left(J^{r,s}{(Y,ℝ)}_0\right)}^{*}$ instead of ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$ are solved.

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.

For natural numbers $r$ and $n$ and a real number $a$ we construct a natural vector bundle $T^{\left(r\right),a}$ over $n$-manifolds such that $T^{\left(r\right),0}$ is the (classical) vector tangent bundle $T^{\left(r\right)}$ of order $r$. For integers $r\ge 1$ and $n\ge 3$ and a real number $a<0$ we classify all natural operators $T_{|M_n}⇝T{T}^{\left(r\right),a}$ lifting vector fields from $n$-manifolds to $T^{\left(r\right),a}$.