Local trivialization of a fiber bundle associated to a principal bundle

Let $$(P,\pi,M,G)$$ be a principal bundle, where $$P$$, $$M$$, $$G$$ are smooth manifolds (total space, base space and fiber=structure group respectively, so $$G$$ is both fiber and structure group)

• the projection map is $$\pi:u\in P\rightarrow p\in M$$ $$\Rightarrow$$ $$\pi^{-1}(U_i)\equiv \{ u\in P: \pi(u)\in U_i \}$$, where $$U_i$$ is an open set in $$M$$
• the local trivialization diffeo is $$\phi_i:(p,g_i)\in U_i\times G\rightarrow \phi_i(p,g_i)\equiv u\in \pi^{-1}(U_i)$$
• $$G$$ acts on the right on $$P$$ globally, $$(u,a)\in P\times G\rightarrow ua =\phi_i(p,g_ia)=\phi_j(p,g_ja)\in P$$ $$\Rightarrow$$ $$\pi(u)=\pi(ua)=p \ \ \forall a \in G$$

Then we can construct a fiber bundle $$(E=P\times F /_G,\pi_E,M,F,G)$$ associated to this principal bundle, which has the same base space $$M$$ and the same structure group $$G$$, a new fiber $$F$$ and

• elements of $$E$$ are the equivalence classes $$[(u,f)]\equiv \{(u',f'): u'=ug, \ f'=g^{-1}f\}$$, where we have used the global right action of $$G$$ on $$P$$ and we have defined a left action of $$G$$ on the new fiber $$F$$
• the projection map is $$\pi_E:[(u,f)] \in E\rightarrow \pi_E([(u,f)])=\pi(u)=p\in M$$

My problem is about the local trivialization of the associated bundle. I know that this diffeo is a map

$$\psi_i: (p,f_i)\in U_i\times F\rightarrow \psi_i(p,f_i)\in\pi_E^{-1}(U_i)$$

but I don't know how to proceed (I'm studying on Nakahara, but this is not very bright on the book). I try to explain myself better: my guess is that $$\psi_i(p,f_i)$$ is the element (=the equivalence class) $$[(u,f_i)]$$, so if we consider another open set $$U_j$$ such that $$U_i \cap U_j \neq \emptyset$$ and a point $$p\in U_i\cap U_j$$, then we can construct the transition function $$h_{ij}:p\in U_i\cap U_j \rightarrow h_{ij}(p)\equiv g \in G$$

$$\psi_i: (p,f_i)\in U_i\cap U_j\times F\rightarrow \psi_i(p,f_i)=[(u,f_i)]\in\pi_E^{-1}(U_i\cap U_j)\\ \psi_j: (p,f_j)\in U_i\cap U_j\times F\rightarrow \psi_j(p,f_j)=[(u,f_j)]\in\pi_E^{-1}(U_i\cap U_j)\\ \Rightarrow h_{ij}(p)\equiv \psi^{-1}_i(p,-)\circ\psi_j(p,-):f_j\in F \rightarrow f_i=h_{ij}(p)f_j=gf_j\in F$$

but from this last relation, it follows that

$$\psi_i(p,gf_j)=\psi_i(p,f_i)=\psi_j(p,f_j) \Rightarrow [(u,gf_j)]=[(u,f_i)]=[(u,f_j)]$$

Is this conclusion correct? Because I don't think it is. So I think that probably I'm interpreting the local trivialization diffeo of an associated bundle incorrectly. Please be indulgent, I'm not well versed in this topic. Thank you.

I think I found an answer to my question: the correct definition of the local trivialization diffeo should be

$$\psi_i : (p,f_i)\in U_i\times F\rightarrow \psi_i(p,f_i)=[(\phi_i(p,e),f_i)]\in\pi_E^{-1}(U_i)$$

where $$e$$ is the identity of the structure group $$G$$ (which is also the fiber of $$P$$). Here we are using the canonical local trivialization $$\phi_i$$ of the principal bundle, so $$\phi_i(p,e)=s_i(p)$$ is the section.

This definition allows us to construct the following transition function

$$h_{ij}(p)\equiv \psi^{-1}_i(p,-)\circ\psi_j(p,-):f_j\in F \rightarrow f_i=h_{ij}(p)f_j=gf_j\in F$$

therefore

$$\psi_i(p,gf_j)=\psi_i(p,f_i)=\psi_j(p,f_j) \Rightarrow [(\phi_i(p,e),gf_j)]=[(\phi_i(p,e),f_i)]=[(\phi_j(p,e),f_j)]$$

but we know that $$t_{ij}(p)=\phi_{i,p}^{-1}\circ \psi_{j,p}$$ and $$t_{ij}(p)\equiv a \in G$$

$$[(\phi_{j,p}(e),f_j)]=[(\phi_{i,p}\circ \phi_{i,p}^{-1}\circ \phi_{j,p}(e),f_j)]=[(\phi_{i,p}\circ t_{ij}(p)e,f_j)]=[(\phi_{i,p}(e)a,f_j)]=[\phi_{i}(p,e),af_j]\\ [(\phi_{j,p}(e),f_j)]=[(\phi_j(p,e),f_j)]=[(\phi_i(p,e),f_i)]=[(\phi_i(p,e),gf_j)]\\ \Rightarrow [\phi_{i}(p,e),af_j]=[(\phi_i(p,e),gf_j)]$$

so we obtain that the transition functions of $$P$$ and $$E$$ are the same $$a=g$$, correctly.