I am sorry if this is too elementary; I had posted it on math.stack but no one answered.

Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a connection on $P$, where $\mathfrak{g}$ is the Lie algebra of $G$. Associated to every connection there is a curvature $F\in\Omega^{2}(P)\otimes\mathfrak{g}$ defined as

$F = DA$

where $D\colon \Omega^{r}(P)\otimes\mathfrak{g}\to \Omega^{r+1}(P)\otimes\mathfrak{g}$ defined as $D\Omega(X_{1},\dots, X_{r+1}) = d\Omega(X^{H}_{1},\dots, X^{H}_{r+1})$. The superscript $H$ denotes the projection to the horizontal distribution given by the connection and $d\colon \colon \Omega^{r}(P)\to \Omega^{r+1}(P)$. Now, $F$ induces (using local sections), a two form

$\mathcal{F}\in\Omega^{2}(M;\mathrm{ad} P)$

taking values in the adjoint bundle of $P$, which is a vector bundle of rank $dim\, \mathfrak{g}$. The two-form $\mathcal{F}$ satisfies

$\mathcal{D}\mathcal{F} = d\mathcal{F} + [\mathcal{A},\mathcal{F}] = 0$

where $\mathcal{A}$ is the local form of the connection one-form $A$. The two-form $\mathcal{F}$ is a particular case of a form taking values in a vector bundle with a connection. For this kind of forms there is a natural derivative $d_{\nabla}$, the exterior covariant derivative. In the case of $\mathcal{F}$ the vector bundle is $\mathrm{ad}\, P$ and the connection $\nabla$ is the induced on the adjoint bundle by $A$. My question is, what is the relation between $\mathcal{D}$ and $d_{\nabla}$? Are they the same?

Thanks.