  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 \(\textrm{C}^*\)-correspondence
\(\mathcal{H}(X,\lambda)\) from \(\textrm{C}^*(G,\alpha)\) to
\(\textrm{C}^*(H,\beta)\). 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
\(\textrm{C}^*\)-algebras form a bicateogry \(\mathfrak{C}\) with
\(\textrm{C}^*\)-correspondences as 1-arrows. The 2-arrows in \(\mathfrak{C}\)
are unitaries of Hilbert \(\textrm{C}^*\)-modules that intertwine the
representations. In this article, we show that a topological correspondence
going to a \(\textrm{C}^*\)-one is a bifunctor~\(\mathfrak{T}\to\mathfrak{C}\).
