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We introduce the notion of the relative Stone-Čech compactification, which is a generalisation of the Stone-Čech compactification.
We show that the relative Stone-Čech compactification is left adjoint to the inclusion functor of the category of proper Hausdorff G0-spaces into the category of G0-spaces, for a locally compact étale groupoid G.
We establish that a groupoid correspondence which acts on a space also acts on the relative Stone-Čech compactification of the space.
The main result shows the existence of a groupoid model for a diagram of proper locally compact groupoid correspondences and shows that it is again a locally compact étale groupoid, i.e., the universal action occurs on a space that is locally compact Hausdorff and proper over G0.