= Solution
Choose local sections related by $s'=su$, where $u:U\to G$. Keeping the same point of the <principal bundle> requires $s\gamma=s'\gamma'$, so $\gamma'=u^{-1}\gamma$. For <reconstruction of a principal connection from local gauge potentials>, the <local principal connection form> transforms as
$$
A'=u^{-1}Au+u^{-1}du.
$$
Using $d(u^{-1})=-u^{-1}(du)u^{-1}$, compute
$$
(\gamma')^{-1}A'\gamma'=\gamma^{-1}A\gamma+\gamma^{-1}(du)u^{-1}\gamma,
$$
and
$$
(\gamma')^{-1}d\gamma'=\gamma^{-1}d\gamma-\gamma^{-1}(du)u^{-1}\gamma.
$$
The inhomogeneous terms cancel. \b[Thus $\boxed{\omega'=\omega}$]: both local expressions define the same one-form on the total space. This is compatibility across trivializations, whereas $R_h^*\omega=\operatorname{Ad}_{h^{-1}}\omega$ describes the separate right-action equivariance of a <principal connection>. In particular a change of trivialization is not an assertion that the local <gauge potential of a principal connection> $A$ itself is unchanged.
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