Let $Y\to M$ be a fibred manifold with $m$-dimensional base and $n$-dimensional fibres and $E\to M$ be a vector bundle with the same base $M$ and with $n$-dimensional fibres (the same $n$). If $m\ge 2$ and $n\ge 3$, we classify all canonical constructions of a classical linear connection $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ on $Y$ from a system $(\Gamma ,\Lambda ,\Phi ,\Delta )$ consisting of a general connection $\Gamma$ on $Y\to M$, a torsion free classical linear connection $\Lambda$ on $M$, a vertical parallelism $\Phi :Y{\times }_{M}E\to VY$ on $Y$ and a linear connection $\Delta$ on $E\to M$. An example of such $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ is the connection $(\Gamma ,\Lambda ,\Phi ,\Delta )$ by I. Kolář.

First we deduce some general results on the covariant form of the natural transformations of Weil functors. Then we discuss several geometric properties of these transformations, special attention being paid to vector bundles and principal bundles.

For every bundle functor we introduce the concept of subordinated functor. Then we describe subordinated functors for fiber product preserving functors defined on the category of fibered manifolds with $m$-dimensional bases and fibered manifold morphisms with local diffeomorphisms as base maps. In this case we also introduce the concept of the underlying functor. We show that there is an affine structure on fiber product preserving functors.

We characterize Weilian prolongations of natural bundles from the viewpoint of certain recent general results. First we describe the iteration $F\left(EM\right)$ of two natural bundles $E$ and $F$. Then we discuss the Weilian prolongation of an arbitrary associated bundle. These two auxiliary results enables us to solve our original problem.

It is well known that the concept of holonomic $r$-jet can be geometrically characterized in terms of the contact of individual curves. However, this is not true for the semiholonomic $r$-jets, [5], [8]. In the present paper, we discuss systematically the semiholonomic case.