I am reading through lecture notes found here and on pg 11 they define a map $\overline{\phi}_{\alpha}:\pi^{-1}(U_{\alpha})\rightarrow G$ by
$\overline{\phi}_{\alpha}(p)=g_{\alpha}(\Phi(p))g_{\alpha}(p)^{-1}$,
where $\Phi:P\rightarrow P$ is a gauge transformation of a principal bundle $\pi:P\rightarrow M$.
Then about halfway down pg 12 the author justifies a line of working by claiming
$g_{\alpha}(p)=\overline{\phi}_{\alpha}(p)g_{\alpha}(q)$,
where $q=\Phi^{-1}(p)$. But this surely must be incorrect since the equality above is equivalent to
$\overline{\phi}_{\alpha}(p)=g_{\alpha}(p)g_{\alpha}(q)^{-1}=g_{\alpha}(\Phi(q))g_{\alpha}(q)^{-1}=\overline{\phi}_{\alpha}(q)$
which is not possible since $\Phi$ is a diffeomorphism?
The reason this worries me is because he uses this relationship in the next line to justify another step. So either I am missing something or the author has a typo and now I need to go and figure out what it is they actually meant.
If anyone could shed some light on this it would be great. Thanks.
Here is what was probably intended. Observe that $\overline{\phi}_{\alpha}(q)=g_{\alpha}(p)g_{\alpha}(q)^{-1}$. Thus $g_{\alpha}(p)=\overline{\phi}_{\alpha}(q)g_{\alpha}(q)$.