The \(r\)-th principal (or gauge-natural) prolongation \(W^{r}P \rightarrow M\) of a principal bundle \(P \rightarrow M\) is a fundamental structure for both the theory of geometric objects fields, and the gauge theories of mathematical physics. In a previous work a principal bundle \(W^{F}P\) for every fiber product preserving bundle functor \(F\) on the category \(\mathcal{F} \mathcal{M}_{m}\) of fibered manifolds with \(m\)-dimensional bases and fibered morphisms with local diffeomorphisms as base maps was constructed.
In the paper under review the author shows that every fiber product preserving bundle functor \(F\) on the category \(\mathcal{F} \mathcal{M}_{m}\) defines the gauge-like prolongation \(W^{F}P\) of a principal bundle \(P\), that coincides with the \(r\)-th principal prolongation \(W^{r}P\) in the special case \(F = J^{r}\). For a large class of such functors the author introduces the torsion of connections of \(W^{F}P\) and obtains some of its properties analogous to the case of \(W^{r}P\).