Fiber product preserving bundle functors of vertical type

We extend the concept of vertical Weil functors VAVA corresponding to Weil algebras A   to the one of generalized vertical Weil functors VAVA on FMmFMm corresponding to Weil algebra bundle functors A   on MfmMfm. Next, we show that the fiber product preserving bundle functors F   on FMmFMm of vertical type are the generalized vertical Weil functors VAVA on FMmFMm. That F   is of vertical type it means that Fxf:FxY→FxY1Fxf:FxY→FxY1 depends only on fx:Yx→Yx1 for any FMmFMm objects Y   and Y1Y1 with the same base and any FMmFMm-map f:Y→Y1f:Y→Y1 with the identity as the base map.