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 ${\text{C}}^{\ast }$-correspondence $\mathcal{H}(X,\lambda )$ from ${\text{C}}^{\ast }(G,\alpha )$ to ${\text{C}}^{\ast }(H,\beta )$. In one of our earlier article we described composition two topological correspondences. In the present article, we prove that second countable locally compact Hausdorff topological groupoids with Haar systems form a bicategory $\mathfrak{T}$ when equipped with a topological correspondences as 1-arrows. The equivariant homeomorphisms of topological correspondences preserving the families of measures are the 2-arrows in~$\mathfrak{T}$. One the other hand, it well-known that ${\text{C}}^{\ast }$-algebras form a bicateogry $\mathfrak{C}$ with ${\text{C}}^{\ast }$-correspondences as 1-arrows. The 2-arrows in $\mathfrak{C}$ are unitaries of Hilbert ${\text{C}}^{\ast }$-modules that intertwine the representations. In this article, we show that a topological correspondence going to a ${\text{C}}^{\ast }$-one is a bifunctor~$\mathfrak{T}\to \mathfrak{C}$.