Some properties of fiber product preserving bundle functors

Let F be a fiber product preserving bundle functor on the category FMm of the proper base order r. We deduce that the r-th principal gauge prolongation WrP of a principal bundle P is a reduction of the principal bundle WFP constructed in Doupovec and Kolář (2001) [1]. In certain special cases, we solve the problem up to what extent F is determined by the underlying group homomorphism.