Non-existence of some canonical constructions on connections

summary:For a vector bundle functor $H:\Cal M f\to \Cal V\Cal B$ with the point property we prove that $H$ is product preserving if and only if for any $m$ and $n$ there is an $\Cal F\Cal M_{m,n}$-natural operator $D$ transforming connections $\Gamma$ on $(m,n)$-dimensional fibered manifolds $p:Y\to M$ into connections $D(\Gamma)$ on $Hp:HY\to HM$. For a bundle functor $E:\Cal F\Cal M_{m,n}\to \Cal F\Cal M$ with some weak conditions we prove non-existence of $\Cal F\Cal M_{m,n}$-natural operators $D$ transforming connections $\Gamma$ on $(m,n)$-dimensional fibered manifolds $Y\to M$ into connections $D(\Gamma)$ on $EY\to M$