On Differential Forms in Functional Spaces of Pre-Phase Space

Question

I am presently reading this [1] paper on covariant phase space and I have difficulty understanding the following formalism developed:

In the paper (section $2.2$, pg. $12$), the authors have introduced the notion of pre-phase space and go on to reinterpret differential forms by their functional counterpart. Instead of viewing $\delta$ as the variation of a functional, it is viewed as an exterior derivative living in the configuration space. Thus, the action of $\delta \phi^{a}$ is given by
$$\delta \phi^{a}\left(\int d^{d}x'f^{b}\left(\phi,x' \right)\frac{\delta}{\delta \phi^{b}(x')} \right)=f^{a}(\phi,x)$$  
They go on to derive a formula for the pre-symplectic current by making the assumption that $\delta^{2}=0$ (which holds since the functional is being viewed as an exterior derivative). Finally, in section $2.3$, they follow this formalism to define a vector field as follows
$$X_{\xi}\equiv\int d^{d}x\mathcal{L}_{\xi}\phi^{a}(x)\frac{\delta}{\delta \phi^{a}}$$
such that $\cdot$ in $X_{\xi}\cdot \delta \phi^{a}(x)$ denotes the insertion of a vector into the first arguement of the differential form.  

I don't follow the formalism used, are they stating that the differential forms have the above-stated form in the functional space? If this is so then how does one prove this and that the assumption $\delta^{2}$ holds. Also, in section $2.2$, a statement is made that it is convenient to re-interpret functionals $\Theta$ (the symplectic potential) and $C$ (a general $(d-2)-$form) as one-forms on the pre-phase space. How is this obvious?  

  [1]: https://arxiv.org/abs/1906.08616