  We prove that the cotangent of a double Lie groupoid S has itself a double
groupoid structure with sides the duals of associated Lie algebroids, and
double base the dual of the Lie algebroid of the core of S. Using this, we
prove a result outlined by Weinstein in 1988, that the side groupoids of a
general symplectic double groupoid are Poisson groupoids in duality. Further,
we prove that any double Lie groupoid gives rise to a pair of Poisson groupoids
(and thus of Lie bialgebroids) in duality. To handle the structures involved
effectively we extend to this context the dualities and canonical isomorphisms
for tangent and cotangent structures of the author and Ping Xu.

Differential Geometry;Symplectic Geometry