In this page [transgression of differential+forms][1] they have at defintion 2.7 they have 

> Let $E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$ be a field bundle over a spacetime $\Sigma$ (def. 2.5), with induced jet bundle $J_{\Sigma}^{\infty}(E)$
For $\Sigma_r \hookrightarrow \Sigma$ be a submanifold of spacetime of dimension $r \in \mathbb{N}$, then transgression of variational differential forms to $\Sigma_r$ is the function
$$
\tau_{\Sigma_r}: \Omega_{\Sigma, \mathrm{cp}}^{r, \bullet}(E) \longrightarrow \Omega^{\bullet}\left(\Gamma_{\Sigma_r}(E)\right)
$$
which sends a differential form $A \in \Omega_{\Sigma, \text { cp }}^{r, \bullet}(E)$ to the differential form $\tau_{\Sigma_r} \in \Omega^{\bullet}\left(\Gamma_{\Sigma_r}(E)\right)$ which to a smooth family on field configurations
$$
\Phi_{(-)}: U \times N_{\Sigma} \Sigma_r \longrightarrow E
$$
assigns the differential form given by first forming the pullback of differential forms along the family of jet prolongation $j_{\Sigma}^{\infty}\left(\Phi_{(-)}\right)$followed by the integration of differential forms over $\Sigma_r$ :
$$
\tau_{\Sigma}A_{\Phi_{(-)}}:=\int_{\Sigma_r}\left(j_{\Sigma}^{\infty}\left(\Phi_{(-)}\right)\right)^*A \in \Omega^{\bullet}(U) .
$$

Why the last expression is a differential form . After integration we have a number  so isn't it a function?

  [1]: https://ncatlab.org/nlab/show/transgression+of+differential+forms