  We introduce the category of set-theoretic representations of a matched pair
of groupoids. This is a monoidal category endowed with a monoidal functor to
the category of quivers over the common base of the groupoids in the matched
pair (the forgetful functor). We study monoidal functors between two such
categories of representations which preserve the forgetful functor. We show
that the centralizer of such a monoidal functor is the category of
representations of a new matched pair, which we construct explicitly. We
introduce the notions of {\em double} of a matched pair of groupoids and {\em
generalized double} of a morphism of matched pairs. We show that the
centralizer of the forgetful functor is the category of representations of the
dual matched pair, and the centralizer of the identity functor (the center) is
the category of representations of the double. We use these constructions to
classify the braidings in the category of representations of a matched pair.
Such braidings are parametrized by certain groupoid-theoretic structures which
we call {\em matched pairs of rotations}.
  Finally, we express our results in terms of the weak Hopf algebra associated
to a matched pair of groupoids. A matched pairs of rotations gives rise to a
quasitriangular structure for the associated weak Hopf algebra. The Drinfeld
double of the weak Hopf algebra of a matched pair is the weak Hopf algebra
associated to the double matched pair.
