  We introduce the notion of a symplectic hopfoid, which is a "groupoid-like"
object in the category of symplectic manifolds where morphisms are given by
canonical relations. Such groupoid-like objects arise when applying a version
of the cotangent functor to the structure maps of a Lie groupoid. We show that
such objects are in one-to-one correspondence with symplectic double groupoids,
generalizing a result of Zakrzewski concerning symplectic double groups and
Hopf algebra objects in the aforementioned category. The proof relies on the
fact that one can realize the core of a symplectic double groupoid as a
symplectic quotient of the total space. The resulting constructions apply more
generally to give a correspondence between double Lie groupoids and
groupoid-like objects in the category of smooth manifolds and smooth relations,
and we show that the cotangent functor relates the two constructions.
