Transition function of a principal bundle (${\Phi }_j{\Phi }_i^{-1}(x,g)=(x,{\psi }_{ji}(x)g)$)

For equivariant local trivializations of a right principal bundle, the change of coordinates is left multiplication in the group coordinate. Indeed an equivariant change $H$ satisfies $H(x,gh)=H(x,g)h$, so evaluating at the identity gives $H(x,g)=(x,\psi (x)g)$. Its inverse shows that $\psi$ takes values in the Lie group, and restriction to $(x,e)$ makes it smooth. Composition on triple overlaps gives ${\psi }_{ki}={\psi }_{kj}{\psi }_{ji}$.