How do we go from a covariant derivative on a principal bundle to a covariant derivative on an associated bundle Let $M$ be a smooth manifold and $\pi:P\to M$ a principal $G$ bundle over $M$. Suppose that $P$ is equipped with a connection one form $\omega$. We can define an exterior covariant derivative on $P$ via $D\eta = d\eta \circ \operatorname{hor}$, where $\operatorname{hor}$ is the horizontal part of any vector in $TP$. For $G$-equivariant functions $\phi :P\to F$ (that is $\phi(p\cdot g) =\rho(g^{-1})\phi(p)$, where $\rho:G\to F$ is a representation of $G$), and indeed for $G$-equivariant forms as well (but we don't need them), the exterior covariant derivative takes the simple form $$D\phi = d \phi + \rho_{*e}(\omega)\wedge \phi,$$where the wedge product can be defined in a natural way. These $G$-equivariant functions are in one-to-one correspondence with local sections of the associated bundle $Q :=P \times _G F$ over an open set $U$ of $M$, and they contain the same information. The question is then, how does one go from the exterior covariant derivative on $G$-equivariant functions to the covariant derivative of local sections of the associated bundle $Q$? One could naturally take the covariant derivative to simply be the local section $\sigma$ associated to $D\phi(X)$, where $X \in TP$, but this is not a covariant derivative on $Q$, since $X$ is not in $TM$. I have also seen the covariant derivative defined by pulling back $D\phi$ by a local section $s$ of $P$ over an open set $U$, but this approach doesn't make much sense to me. This approach would make more sense if the correspondence mentioned above was between global sections and $G$-equivariant functions, in which case we just take the local representation of the section (i.e. $s^*(D\phi)$). If anyone could clarify, it would be very helpful.
 A: In the description of a principal connection via a connection form, this is not so easy to see. The point is that the combination you use to define $D\phi$ is chosen in such a way that it vanishes on any fundamental vector field (if $\phi$ is a form rather than a function, you have to assume  that $\phi$ itself is horizontal). Hence $D\phi(X)$ depends only on the projection of $X$ to $TM$. Otherwise put, for a vector field on $M$, you can choose any lift to a $G$-invariant vector fields on $P$ and insert this into to $D\phi$ to get the function corresponding to the covariant derivative of the section determined by $\phi$. This also leads to a description in terms of local sections.
Alternatively, you can move towards the description of principal connections as distributions. Any tangent vector $X$ on $M$ can be uniquely lifted to tangent vectors along the appropriate fiber of $P$ which are horizontal in the sense that they are annihilated by $\omega$. Applying this point-wise to a smooth vector field on $M$, one obtains a horizontal vector field on $P$ which in addition is automatically $G$-invariant. Using this to differentiate $\phi$ directly produces the equivariant function corresponding to the covariant derivative of $s$ in the direction of the initial vector field.
A: I want to just add more detail to one sentence of Andreas' (correct) answer.
The reason $D\phi(X)$ only depends on the projection of $X\in TP$ to $TM$ is because of the horizontal projection in your definition of $D$.
If $X'$ is any other lift of $\pi(X)$, then $\pi(X-X') = \pi(X)-\pi(X) = 0$ so $X-X'$ is vertical. Then
$$D\phi(X)-D\phi(X') = D\phi(X-X') = d\phi\circ\text{hor}(X-X') = 0.$$
