I'm trying to prove the reconstruction theorem for general smooth fiber bundles, that is, no extra algebraic structure over my bundle, only smoothness.
Let $M$ and $F$ be smooth manifolds of dimensions $n$ and $k,$ respectively, and $\{U_\alpha\}_{\alpha \in A}$ a open cover for $M.$ Suppose for each $\alpha, \beta \in A$ there are maps $\psi_{\alpha\beta} : U_\alpha \cap U_\beta \to Diff(F)$ satisfying the cocycle conditions. Then there exists a smooth fiber bundle $E$ with fiber $F$ and base space $M$ such that the transition functions for local trivializations are $\psi_{\alpha \beta}.$
There are many similar questions about it in this website but I couldn't find how to construct the smooth structure after we construct the set $E$ via the usual equivalence relation. For the sake of completeness, and since absolutely no book or article I've searched so far does this crucial step, here is what I tried so far:
We define $$E = \left(\bigsqcup_\alpha U_\alpha \times F \right) / \sim $$
where $(\alpha,p, f) \sim (\beta,q,g)$ iff $p = q \in U_\alpha \cap U_\beta$ and $f = \psi_{\alpha\beta}(q)(g).$ With a bit of work it can be shown that $\pi : E \to M,$ $\pi([\alpha,p, f]) = p$ and $\Phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times F,$ $\Phi_\alpha([\alpha,p, f]) = (p,f) $ are well defined, $\pi$ is surjective, $\Phi_\alpha$ is invertible, $\pi = \pi_1 \circ \Phi_\alpha$ and $\Phi_\alpha \circ \Phi_\beta ^{-1}(p,f) = (p, \psi_{\alpha\beta}(p)(f)),$ so if I find a smooth structure for $E$ the proof is mostly done, only needing to check the smoothness of those functions. Currently, for the sake of simplicity I'm assuming that there are smooth charts $\varphi_\alpha$ for $M$ defined over each $U_\alpha.$
For this I'm trying to use the usual theorem where maps from a cover of $E$ to open sets of $\mathbb R ^{n+k}$ define a unique topology and smooth structure for $E$ (see for example lemma 1.35 of Lee, introduction to smooth manifolds, 2ed, for the precise wording and notation).
Now the tricky part that halted my progress is that, unlike the similar proof for vector bundles, we need to introduce a atlas for $F,$ and don't know how to proceed with this extra step. If $\{(V_i,\sigma_i)\}_{i \in I}$ is a smooth atlas for $F,$ By analogy with the vector bundle case I'm trying to see of the following family of functions will do the trick:
$$ F_{\alpha,i} : \Phi_\alpha^{-1}({U_\alpha \times V_i}) \to \varphi_\alpha(U_\alpha) \times \sigma(V_i) $$
given by $F_{\alpha,i}([\alpha, p, f]) = (\varphi_\alpha \times \sigma_i)\circ \Phi_\alpha ([\alpha, p, f]) = (\varphi_\alpha(p),\sigma_i(f))$
They are bijective (if we restrict $\Phi_\alpha$) their images are open sets of $\mathbb R^{n+k}.$ By denoting $\Phi_\alpha^{-1}({U_\alpha \times V_i}) = W_{\alpha,i}, $ since both $M$ and $F$ are smooth manifolds we can extract a countable cover for $E$ of the collection $\{W_{\alpha,i}\}.$ Now in accordance to the theorem I cited, I need to check that each $F_{\alpha,i}(W_{\alpha,i}\cap W_{\beta,j})$ is an open set of $\mathbb R^{n+k},$ and I'm stuck here, since it doesn't seems easy to compute the above set. Even if for a moment I suppose they are open, I then need to check that the transition is smooth, but (where they are defined)
$$ F_{\beta,j}\circ F_{\alpha,i}^{-1}(x,y) = (\varphi_\beta \circ \varphi_\alpha^{-1}(x), \sigma_j\Big(\psi_{\beta\alpha}\big(\varphi_{\alpha}^{-1}(x)\big)(\sigma_i^{-1}(y))\Big)) $$ and the second component hardly seems to be smooth.
Maybe these maps are the wrong way to do, but I can't see an alternative.
An my going the right way? are there a better method to prove this last step?
Thanks.