Let $\pi:P\to M$ be a smooth principal right $G$-bundle, where $G$ is a Lie group. Let $\operatorname{Gau}(P)$ denote the group of smooth principal bundle automorphisms $\Phi:P\to P$ covering $\operatorname{id}_M$, meaning $\pi\circ \Phi=\pi$ and $\Phi(pg)=\Phi(p)g$ for all $p\in P$ and $g\in G$.
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Then the assignment sending $\Phi\in \operatorname{Gau}(P)$ to the unique smooth map $\gamma_\Phi:P\to G$ determined by
Under this bijection, if $\Phi_i(p)=p\gamma_i(p)$ for $i\in\{1,2\}$, then $\Phi_2\circ \Phi_1$ corresponds to the map $\gamma_2\star \gamma_1:P\to G$ given by