Natural lifting of connections to vertical bundles

summary:One studies the flow prolongation of projectable vector fields with respect to a bundle functor of order $(r,s,q)$ on the category of fibered manifolds. As a result, one constructs an operator transforming connections on a fibered manifold $Y$ into connections on an arbitrary vertical bundle over $Y$. It is deduced that this operator is the only natural one of finite order and one presents a condition on vertical bundles over $Y$ under which every natural operator in question has finite order