The bicategory of topological correspondences

It is known that a topological correspondence (X,\lambda) from a locally compact groupoid with a Haar system (G,\alpha) to another one, (H,\beta), produces a C*-correspondence H(X,\lambda) from C^*(G,\alpha) to C^*(H,\beta). We described the composition of two topological correspondences in one of our earlier articles. In the present article, we prove that second countable locally compact Hausdorff groupoids with Haar systems form a bicategory T when equipped with topological correspondences as 1-arrows and isomorphisms of topological correspondences as 2-arrows. On the other hand, it well-known that C*-algebras form a bicategory C with C*-correspondences as 1-arrows, and the unitary isomorphisms of Hilbert C*-modules that intertwine the representations serve as the 2-arrows. In this article, we show that a topological correspondence going to a C*-one is a bifunctor T to C. Finally, we show that in the sub-bicategory of T consisting of the Macho-Stadler-O'uchi correspondences, invertible 1-arrows are exactly the groupoid equivalences.