I'm trying to understand the proof of the Ambrose-Singer theorem, following, for instance, Werner Ballmann's notes. I understand that the gist of the theorem is that, when you have a vector bundle $(E,\nabla) \to M$, the holonomy algebra $\mathfrak{hol}_x(\nabla)$ is going to be generated by conjugates of curvature operators $R^\nabla(v,w)$, $v,w \in T_xM$, by parallel transport operators $P_\alpha\colon E_x \to E_x$, where $\alpha \in \Omega(M,x)$ is a loop. In more than one place, I have seen the following notation
$$R^\nabla_\alpha (v,w) \doteq P_\alpha \circ R^\nabla({\color{red}{P_\alpha v}},{\color{red}{P_\alpha w}}) \circ P_{\alpha}^{-1}.$$Computing $P_\alpha$ at $v$ and $w$ makes absolutely no sense to me. They live in $T_xM$, not in $E_x$! The only thing I can imagine is that since $\nabla$ induces a horizontal distribution on the total space $E$ (namely, one sets $v^{\sf h}_{(x,\phi)} \in E_x$ to be ${\rm d}\psi_x(v)$, where $\psi$ is any local section around $x$ with $(\nabla\psi)_x=0$ and $\psi_x=\phi$), one would mean to apply $P_\alpha$ to horizontal lifts -- and even still -- the matter of the "height" $\phi \in E_x$ remains undecided. Not to say nothing about horizontal lifts is said anywhere. And I don't think you're supposed to arbitrarily pick an auxilliary connection in $TM$ to do it.
What is going on?