Let $P \rightarrow M$ be a smooth principal $G$-bundle. Notation: use $\Gamma$ to denote global (smooth) sections of a bundle. Also, $G$ acts on the left on $P$.
I can't seem to find a specific definition of a covariant derivative on $P$ (though definitions of connections abound), so I was wondering if the following definition makes sense?
Definition: Let's say a function $\nabla : \Gamma(TM) \times \Gamma(P) \rightarrow C^{\infty}(M\to\mathrm{Lie}G)$, written $(X,s) \mapsto \nabla_X s$, is a covariant derivative on $P$ if:
(1) $\nabla$ is $C^{\infty}(M,\mathbb{R})$-linear in the $\Gamma(TM)$ argument, i.e. $\nabla_X s$ is $C^\infty(M)$-linear in $X$; and
(2) for $g : M \rightarrow G$ smooth, for $s \in \Gamma(TM)$, for $X \in \Gamma(TM)$, we have $$ \nabla_X(gs) = \mathrm{Ad}_g(\nabla_X s) + \theta_G(dg(X)) $$ where $\theta_G$ is the Maurer-Cartan form on $G$.
Would the above be the right definition of a covariant derivative on $P$? Is there a source which describes this in detail?