In this post Observables By Urs Schreiber he denotes the space of distributional sections
let $\Gamma_{\Sigma}^{\prime}\left(E^*\right):=\left(\Gamma_{\Sigma, c p}(E)\right)^*$ in the definition 7.9 That is if $u \in \Gamma_{\Sigma}^{\prime}\left(E^*\right) $ than $u$ is a linear functional that takes as argument sections of a vector bundle $E$
$ u_{(b)} \in \Gamma_{\Sigma, s}^{\prime}(E^*)$
In the same post he has proposition 7.10
Let $E \stackrel{f b}{\rightarrow} \Sigma$ be a smooth vector bundle over Minkowski spacetime and let $s \in\{c p, \pm c p, s c p, t c p\}$ be any of the support conditions from def. $2.36$. Then the operation of regarding a compactly supported smooth section of the dual vector bundle as a functional on sections with this support property is a dense subspace inclusion into the topological vector space of distributional sections from def. 7.9: $$ \begin{array}{ccc} \Gamma_{\Sigma, \mathrm{cp}}\left(E^*\right) & \stackrel{u_{(-)}}{\longrightarrow} & \Gamma_{\Sigma, S}^{\prime}(E) \\ b & \mapsto & \left(\Phi \mapsto \int_{\Sigma} b(x) \cdot \Phi(x) \operatorname{dvol}_{\Sigma}(x)\right) \end{array} $$
In my understanding $u_{()}$ is a map from the space of sections of the dual bundle to the space of the distributional section .
Why $ u_{()} \in \Gamma_{\Sigma, s}^{\prime}(E)$ ? Shouldn't we have $ u_{()} \in \Gamma_{\Sigma, s}^{\prime}(E^*)$
To me this appears as a typo, $\Gamma_{\Sigma,s}'(E)$ is not even defined around there and if the author means something else like $\Gamma_{\Sigma,s}'((E^*)^*)$ I would at least have expected a note.
The space $(\Gamma_{\Sigma,cp}(E))^*$ may be seen as "distributions on sections in $E$", suggesting the notation $\Gamma_{\Sigma}'(E)$, or as sections in $E^*$ which are not smooth, but "at most distributional", suggesting to denote it $\Gamma_{\Sigma}'(E^*)$. I guess the author just mixed them up.