  We use symplectic reduction to give a new construction of the core $C$ of a
symplectic double groupoid $D$ as the common leaf space of characteristic
foliations associated to various coisotropic submanifolds of $D$. In the case
of the cotangent double groupoid of a Lie groupoid $G$, the canonical relations
arising from this process turn out to be cotangent lifts of structure maps
associated to $G$. We also show that under this reduction procedure the double
groupoid structure on $D$ descends to a groupoid structure on the leaf space
above, recovering the core groupoid structure on $C$ of Brown and Mackenzie.
