  In the setting of von Neumann algebras, measurable quantum groupoids have
successfully been axiomatized and studied by Enock, Vallin, and Lesieur,
whereas in the setting of $C^{*}$-algebras, a similar theory of locally compact
quantum groupoids could not yet be developed. Some basic building blocks for
such a theory, like analogues of a Hopf-von Neumann bimodule and of a
pseudo-multiplicative unitary, were introduced in the thesis and a recent
article by the author. That approach, however, is restricted to decomposable
quantum groupoids which generalize $r$-discrete groupoids. Recently, we
developed a general approach that covers all locally compact groupoids. In this
article, we explain how the special theory of our thesis embeds into the
general one.
