Sourau's definition of diffeology may be phrased as defining a concrete sheaf on the category $\mathsf{Open}$ of open subsets of Euclidean/coordinate spaces. It seems to me, unless I am missing something, that any such sheaf extends to a sheaf on the site $\mathsf{Man}$ of all smooth manifolds. Is there a proof written down somewhere?

More generally it seems to me that the inclusion $\mathsf{Open} \hookrightarrow \mathsf{Man}$ should induce an equivalence of 2-categories between the 2-category of stacks on $\mathsf{Man}$ and the stacks on $\mathsf{Open}$. Is this true? If so, is there a reference? (Metzler seems to mentions something like this in passing in here: arXiv:math/0306176 [math.DG] .)

Souriau's definition of diffeology may be phrased as defining a concrete sheaf on the category $\mathsf{Open}$ of open subsets of Euclidean/coordinate spaces. It seems to me, unless I am missing something, that any such sheaf extends to a sheaf on the site $\mathsf{Man}$ of all smooth manifolds. Is there a proof written down somewhere?

More generally it seems to me that the inclusion $\mathsf{Open} \hookrightarrow \mathsf{Man}$ should induce an equivalence of 2-categories between the 2-category of stacks on $\mathsf{Man}$ and the stacks on $\mathsf{Open}$. Is this true? If so, is there a reference? (Metzler seems to mentions something like this in passing in here: arXiv:math/0306176 [math.DG] .)