In the The principal connection for a connection form section of Wikipdia Connection form article, first there is a local trivialization $\{U_k,φ_k\}$ with structure group $G$ and transiton functions $\{h_{ij}\}$ of vector bundle $(E,π,M)$, choose local $G$-frames $\{e_k\}$ which are related by $e_V=e_U\cdot h_{UV}$.
Then construct the associated principal $G$-bundle $F_GE=\bigsqcup_U U\times G/\sim$ where the equivalence relation $\sim$ is defined by:
$$((x,g_U)\in U\times G)\sim((x,g_V)\in V\times G) \iff g_U=h_{UV}g_V$$
(note: in the article it is $g_U=h_{UV}^{-1} g_V$ which is against the transition function tradition and also prevent to form the global form later so should be a bug)
Then construct a form on each $U\times G$:
$$ω_{U(x,g)}=Ad_{g^{-1}}π_1^*ω(e_U)+π_2^*ω_{\mathfrak g}$$
where $π_1$ and $π_2$ are projections from $U\times G$ to $U$ and $G$ respectively , $ω_{\mathfrak g}$ is the Maurer-Cartan form, $ω(e_U)$ is the local connection form on $U$ by local section $e_U$.
Then prove that these forms are compatible with each other on the intersections $U\cap V$ then equivalently a form on $F_GE$ is constructed.
But the problem is, to construct a form on $F_GE$, it is not enough to just define forms $ω_{U(x,g)}$ on $U\times G$, there need to be one more step. One method of such step is:
$$η_U=φ_U^*ω_U$$
that is, pullback one more time by the trivialization map $φ_U$, and the purpose is to prove that $η_U=η_V$ on $π^{-1}(U\cap V)$.
The problem of this method is that, suppose $p\in F_GE$, $π(p)\in U\cap V$, $φ_U(p)=(x,g_U)$, $φ_V(p)=(x,g_V)$, then when some tangent vector $X$ in $T_pF_GE$ is applied to $η_U$ and $η_V$, through the pushforward $φ_{U*}$ and $φ_{V*}$ it become different tangent vectors $X_U\in T_{(x,g_U)}(U\times G)$ and $X_V\in T_{(x,g_V)}(V\times G)$. Since $X_U$ and $X_V$ don't have any useful relations, that's a dead end. Then how to even define the "contruction" properly?