Vertical functors on fibered manifolds

The usual vertical functor V on all fibered manifold morphisms is a classical example of a fiber product preserving bundle functor. In the present paper we extend almost directly the concept of the vertical functor V to the ones of modified or generalized vertical Weil functors and using them we describe completely the fiber product preserving bundle functors on many categories of fibered manifolds. More precisely, we fix an admissible category C over manifolds and let FMC be the category of fibered manifolds with bases being C-objects and their fibered maps with base maps being C-morphisms. We introduce the concept of modified vertical Weil functor on FMC and prove that any finite order fiber product preserving regular bundle functor on FMC is isomorphic to a modified vertical Weil functor. We also study finite order regular fiber product preserving bundle functors of vertical type and describe these functors by generalized vertical Weil functors. Finally we study the iteration of finite order regular fiber product preserving bundle functors on FMC.