Let $E \rightarrow X$ be a smooth vector bundle, with jet bundle $J^\infty_X(E) \rightarrow X$.
I am reading Differential Cohomology by Schreiber (link; it is "v2" here), and on page 6 just above equation (1.1), the author says that a base-$X$ smooth vector bundle morphism $L : J_X^\infty(E) \rightarrow \Lambda^{p}(T^*X)$ may be viewed as a differential form on $J_X^\infty(E)$. However, the author doesn't explain this in detail.
Would anyone be able to help explain how we can view $L$ as a differential form on the jet bundle $J_X^\infty(E)$? Any help would be much appreciated!
Note: the author uses the notation $E\rightarrow \Lambda^p(T^*X)$ to mean an actual vector bundle morphism $J_X^\infty(E) \rightarrow \Lambda^p(T^*X)$. Also, the author uses $\Sigma$ where I use $X$.