On the flow prolongation of vector fields

Let $F$ be a fiber product preserving bundle functor of the base order $r$ on the category $\mathcal{FM}_m$ and $\eta$ a projectable vector field on $Y\to M$ over a vector field $\xi$ on $M,m=\dim M$. We construct a natural map transforming $F\eta$ and $j^r\xi$ into the flow prolongation of $\eta$ and deduce its basic properties. Our main tool is a similar construction in the case of products of two manifolds and of product vector fields.