Let $P=(P,\pi,M,G)$ be a principal fibre bundle and $\omega$ a principal connection on it. If $\lambda:G\times S\rightarrow S$ is a smooth left action of $G$ on a manifold $S$, the associated fibre bundle $P_{\lambda,S}$ may be constructed, and as it is well-known, there is a canonically induced connection $\omega_{\lambda,S}$ on $P_{\lambda,S}$.
Now the question is the following. Assuming $P$ and $\omega$ are given as above, $\Gamma$ is a symmetric (torsionless) principal connection on $LM$ (the bundle of linear frames over $M$), and $W^{r,s}P$ is the order $(r,s)$ gauge-natural prolongation of $P$, is there a unique principal connection induced on $W^{r,s}P$ from the data of $(\Gamma,\omega)$?
Intuitively, I'd say yes, and I seem to remember reading about it, but flipping through KMS (Natural Operations in Differential Geometry) I did not seem to find anything about this, so I might have remembered wrong.
A particular case I am interested in is the following: The affine bundle $\mathcal C(P)\rightarrow M$ of principal connections on $P$ is a $W^{1,1}P$-associated bundle. If $(\Gamma,\omega)$ induces a unique connection on $\mathcal C(P)$, then it becomes possible to take the covariant derivative of a principal connection with respect to itself, although the symmetric frame connection $\Gamma$ might be needed as an extra data. I am interested in describing this covariant derivative explicitly in terms of the local connection forms/coefficients.
The following heuristic reasoning seems to imply that such a lift (at least to $\mathcal C(P)$ if not generally) exists, is independent of $\Gamma$ and the covariant derivative of a connection with respect to itself can be interpreted as the curvature form of the connection.
Specifically for $E=P_{\rho, V}$ an associated vector bundle, the covariant derivative can be formally determined from a very "classical" point of view as follows. If $\phi=(\phi^i)$ is the representation of a section of $E$ over a trivialization domain $U\subseteq M$, under a change of trivialization (assumed to be over the same domain for simplicity) affects $\phi$ as $$ \bar\phi=\rho(g^{-1})\phi,\quad g:U\rightarrow G. $$ In the case when $g$ is "infinitesimally close to the identity", we have $g=1+\varepsilon X$, where $X$ is $\mathfrak g$-valued and then $$ \bar\phi=\phi-\varepsilon\rho(X)\phi $$(in $\rho(X)$, $\rho$ is the induced Lie algebra representation), thus the "variation" of $\phi$ under an infinitesimal gauge transformation is $$ \delta\phi=-\rho(X)\phi. $$
Then if a local connection $1$-form $\omega_\mu$ (matrix Lie algebra valued for simplicity; $x=(x^\mu)$ is a coordinate system on $U$), we define $$ \delta_\mu\phi:=-\rho(\omega_\mu)\phi $$, i.e. substitute $\omega_\mu$ for $X$ and the covariant derivative by $$ D_\mu\phi=\partial_\mu\phi-\delta_\mu\phi=\partial_\mu\phi+\rho(\omega_\mu)\phi. $$
Doing this with the connection itself, as $\bar\omega_\mu=g^{-1}\omega_\mu g+g^{-1}\partial_\mu g$, infinitesimally $\delta\omega_\nu=\partial_\nu X+[\omega_\nu,X]$, therefore $$ D_\mu\omega_\nu=\partial_\mu\omega_\nu-\delta_\mu\omega_\nu=\partial_\mu\omega_\nu-\partial_\nu\omega_\mu-[\omega_\nu,\omega_\mu] =\Omega_{\mu\nu}. $$
I expect that the hypothetical gauge-natural lift of $(\Gamma,\omega)$ to $\mathcal C(P)$ is independent of $\Gamma$ and reproduces this calculation.
