The de Rham complex can be viewed as a sheaf $\Omega$ on the entire site $\mathsf{Man}$ of smooth manifolds via the usual pullback of differential forms $(f: M \to N) \mapsto (f^*: \Omega(N) \to \Omega(M))$. So, the de Rham complex is a smooth space. Diffeological spaces, for example, are concrete sheaves on the site of smooth manifolds, but I don't know enough about sheaves to say whether or not the de Rham complex is a diffeological space.

We can also consider the category of open sets of $M$ as a site $O(M)$, where the arrows are inclusions. Then we have the sheaf $\Omega_M \in \mathsf{Sh}(M)$ of differential forms on $M$. Likewise, we have the sheaf $\Omega_N \in \mathsf{Sh}(N)$ of differential forms on $N$.

Now, as I understand it, there are morphisms $i_M: \mathsf{Man} \to M$ and $i_N: \mathsf{Man} \to N$ of sites, given by the respective inclusion functors $i_M^t: O(M) \to \mathsf{Man}$ and $i_N^t: O(N) \to \mathsf{Man}$.

Can we describe the sheaves of differential forms $\Omega_M$ and $\Omega_N$ on $M$ and $N$ in terms of these morphisms of sites? I figure they should somehow be induced by the sheaf $\Omega$ on all of $\mathsf{Man}$, but I am not very experienced with sheaves on sites yet. Moreover, given a smooth map $f: M \to N$, there should a relationship between the inverse image functor and the pullback of forms.