Let $W_m^{r}P$ be a principal prolongation of a principal bundle P → M. We classify all gauge natural operators transforming principal connections on P → M and rth order linear connections on M into general connections on $W_m^{r}P\to M$. We also describe all geometric constructions of classical linear connections on $W_m^{r}P$ from principal connections on P → M and rth order linear connections on M.

The main result is the classification of all gauge bundle functors H on the category ${ℬ}_m\left(G\right)$ which admit gauge natural operators transforming principal connections on P → M into general connections on HP → M. We also describe all gauge natural operators of this type. Similar problems are solved for the prolongation of principal connections to HP → P. A special attention is paid to linear connections.