Fiber product preserving bundle functors on fibered-fibered manifolds

We introduce the concept of modified vertical Weil functors on the category $\F_2\M_{m_1,m_2}$ of fibered-fibered manifolds with $(m_1,m_2)$-dimensional bases and their local fibered-fibered maps with local fibered diffeomorphisms as base maps. We then describe all fiber product preserving bundle functors on $\F_2\M_{m_1,m_2}$ in terms of modified vertical Weil functors.