We generalize the concept of an $(r,s,q)$-jet to the concept of a non-holonomic $(r,s,q)$-jet. We define the composition of such objects and introduce a bundle functor ${\tilde{J}}^{r,s,q}ℱ{ℳ}_{k,l}\times ℱℳ$ defined on the product category of $(k,l)$-dimensional fibered manifolds with local fibered isomorphisms and the category of fibered manifolds with fibered maps. We give the description of such functors from the point of view of the theory of Weil functors. Further, we introduce a bundle functor ${\tilde{J}}_1^{r,s,q}2\text{-}ℱ{ℳ}_{k,l}\to ℱℳ$ defined on the category of $2$-fibered manifolds with $ℱ{ℳ}_{k,l}$-underlying...