Algebraic quantum groupoids have been developed by two of the authors (AVD
and SHW) of this note in a series of papers. Regular multiplier Hopf algebroids
are obtained also by two authors (TT and AVD). Integral theory and duality for
those have been studied by one author here (TT). Finally, again two authors of
us (TT and AVD) have investigated the relation between weak multiplier Hopf
algebras and multiplier Hopf algebroids. In the paper 'Weak multiplier Hopf
algebras III. Integrals and duality' (by AVD and SHW), one of the main results
is that the dual of an algebraic quantum groupoid, admits a dual of the same
type. In the paper 'On duality of algebraic quantum groupoids' (by TT), a
result of the same nature is obtained for regular multiplier Hopf algebroids
with a single faithful integral. The duality of regular weak multiplier Hopf
algebras with a single integral can be obtained from the duality of regular
multiplier Hopf algebroids. That is however not the obvious way to obtain this
result. It is more difficult and less natural than the direct way. We will
discuss this statement further in the paper. Nevertheless, it is interesting to
investigate the relation between the two approaches to duality in greater
detail. This is what we do in this paper. We build further on the intimate
relation between weak multiplier Hopf algebras and multiplier Hopf algebroids.
We now add the presence of integrals. That seems to be done best in a framework
of dual pairs. It is in fact more general than the duality of these objects
coming with integrals