I'm following Sharpe's "Differential Geometry: Cartan's Generalization of Klein's Erlangen Program" book and I have trouble comparing some notions (as introduced in this book) with the way I've learned them in the past. Sharpe says that if the model geometry is reductive, then the universal covariant derivative decomposes as $\tilde D_X=\tilde D_{\mathfrak hX}+\tilde D_{\mathfrak pX}$, where $\mathfrak p=\mathfrak {g/h}$ and $\mathfrak g=\mathfrak h+\mathfrak p$. We see that $\tilde D_{\mathfrak h}$ simply shows the way a function $f\in \Omega^0(P)\otimes (V,\rho)$ (a tensor of type $(V,\rho)$ or a function on the total space of a principal $H$-bundle, i.e the Cartan geometry $(P,\omega)$) transforms under representations of the structure group $H$, specifically $\tilde D_{\mathfrak h}=-\rho_*$. Sharpe says that the projection $\tilde D_{\mathfrak p}$ is the usual covariant derivative $D$ (some denote it as $\nabla$), for which it holds that $$D_X\phi=X(\phi)-\rho_*((\sigma^*\omega_{\mathfrak h})(X))\phi$$ for $X\in\Gamma(TU)$. Here, $U$ is a small open subset of the base space $M$, $\rho_*$ is the push-forward of the representation map $\rho:H\to GL(V)$ and $V$ the representation space, where $F=(V,\rho)$ should be the fibre of the associated vector bundle with total space $(P\times (V,\rho))/H$. Moreover, $\phi=\Phi\circ\sigma$ is the expression of a tensor $\Phi$ of type $(V,\rho)$ in the Cartan gauge for the section $\sigma:U\to\pi^{-1}(U)$. He later says that $\omega_{\mathfrak h}$ is an Ehresmann connection, so I start comparing with what I know.
Given a principal $H$-bundle, an Ehresmann connection $\tilde\omega\in\Omega^1(P)\otimes \mathfrak h$ is a map $\tilde\omega:TP\to\mathfrak h$ with the usual connection properties. We can build a covariant derivative, which has the usual properties and for which it holds that $$D_X\phi=d\phi(X)+(\sigma^*\tilde\omega)(X)\phi$$
I really cannot see how the 2 relations are the same. Perhaps the second definition is invalid? In the first one, $\rho_*((\sigma^*\omega_{\mathfrak h})(X))\in \mathfrak{gl}(F)=End(F)$, so it is expecting a tensor of type $(V,\rho)$ and it gets one. The result lies again in $F$ as expected. Furthermore, $X(\phi)$ should indeed have values in $F$. $D_X\phi$ should have values in $F$, since $D\phi\in\Omega^1(U)\otimes F$ is an F-valued 1-form. I really don't understand how this is valid, when we use $V$ instead of $F=(V,\rho)$ (as Sharpe does), since then we have $\rho_*((\sigma^*\omega_{\mathfrak h})(X))\in \mathfrak{gl}(V)=End(V)$ and this is expecting a vector of $V$. On the other hand, in the last relation, $(\sigma^*\tilde\omega)(X)\in\mathfrak h$, so $(\sigma^*\tilde\omega)(X)\phi$ and $d\phi(X)$ (as pairing product) should have values in $F$ (not sure about that)? Also, shouldn't $D_X\phi$ be again a section of the assciated vector bundle?
I would appreciate any guidance, cause I'm really stuck and confused with this matter. Also, on the whole procedure of the covariant derivative construction, Sharpe does not speak of $H-equivariance$ at all. Let me know if any further clarifications are needed.