The following is a theorem cited as a result of Weinstein in McDuff & Salamon's Introduction to Symplectic Topology, Section 6.2.
Theorem: let $G \rightarrow Symp(F,\sigma)$ be a Hamiltonian action. Then every connection on a principal $G$-bundle $P\rightarrow B$ gives rise to a closed 2-form $\tau$ on the associated fibration $P \times_G F \rightarrow B$ which restricts to the forms $\sigma_b$ on the fibres.
The proof starts by considering the easier case when one takes $F = T^*G$, which is in a natural way a symplectic manifold endowed with a Hamiltonian (left) action of $G$. I can follow the idea of the proof besides one technicality, which is claiming that the associated fiber bundle $P \times_G T^*G$ can be identified with the vertical cotangent bundle of $P$. I think this is wrong, or at least that some pullback is missing: the vertical cotangent bundle of $P$ is a bundle over $P$ (not over $B$, as the proof claims!), while $P\times_G T^*G$ is indeed a bundle over $B$.
The rest of the proof (at least in this easier case) is more or less straightforward once one understands connections on principal bundles in terms of horizontal distributions or splittings of the short exact sequence $$ 0 \rightarrow Vert(P) \rightarrow TP \rightarrow \pi^*TM \rightarrow 0. $$
I am looking for either a detailed explanation of how the pullback is missing and how to fix the proof or an entirely different proof of this fact.
For sake of completeness: the proof proceeds arguing that once one chooses a connection, i.e. a surjection $A: TP \rightarrow Vert(P)$ one can dualize to obtain a $G$-equivariant injection $i: Vert(P)^* \rightarrow T^* P$ from which one can pull the canonical symplectic form of $T^* P$ to $Vert(P)^*$. By virtue of the claim that I dispute, this would yield the desired closed 2-form on $P\times_G T^*G$.