Suppose $\pi:P\to X$ is a principal $G$-bundle with a connection $1$-form $\omega$, $\rho:G\to \text{GL}(V)$ is a linear representation (where $V$ is a finite-dimensional real vector space), and $E:=P\times_\rho V$ is the associated vector bundle. In section 3.2 of Morgan's book on Seiberg-Witten equations, there are given two different definitions of a covariant derivative $\nabla:\Omega^0(X;E)\to \Omega^1(X;E)$. Here are the two definitions:
For a section $\sigma \in \Omega^0(X;E)$, $x\in X$, and $\tau\in T_xX$, choose a path $\gamma$ in $X$ with $\gamma(0)=x$ and $\gamma'(0)=\tau$, and choose a horizontal lift $\tilde{\gamma}$ to $E$. Then $\sigma(\gamma(t))=[\tilde{\gamma}(t),v(t)]\in E$ for some smooth path $v(t)\in V$. Define $\nabla(\sigma)(\tau)=[\tilde{\gamma}(0),v'(0)]\in E$.
Consider the differential $d\rho: \mathfrak{g}\to \text{End}(V)$. Fix a local trivialization $E|_U=U\times V$ and write $\sigma(u)=(u,\alpha(u))$, and define $\nabla(\sigma)=d\rho(\omega)(\alpha)+d\alpha$.
Why are these two definitions equivalent? I can't see how to show it, and I couldn't find any references concerning theses definitions (I've looked Lee's and Tu's books in DG).