  Quasifolds are spaces that are locally modelled by quotients of
$\mathbb{R}^n$ by countable affine group actions. These spaces first appeared
in Elisa Prato's generalization of the Delzant construction, and special cases
include leaf spaces of irrational linear flows on the torus, and orbifolds. We
consider the category of diffeological quasifolds, which embeds in the category
of diffeological spaces, and the bicategory of quasifold groupoids, which
embeds in the bicategory of Lie groupoids, bibundles, and bibundle morphisms.
We prove that, restricting to those morphisms that are locally invertible, and
to quasifold groupoids that are effective, the functor taking a quasifold
groupoid to its diffeological orbit space is an equivalence of the underlying
categories. These results complete and extend earlier work with Masrour Zoghi.
