Double Lie algebroids and second-order geometry, II

We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We show that the Lie algebroid structure of an LA–groupoid may be prolonged to the Lie algebroid of its Lie groupoid structure; in the case of a double groupoid this prolonged structure for either LA–groupoid is canonically isomorphic to the Lie algebroid structure associated with the other; this extends many canonical isomorphisms associated with iterated tangent and cotangent structures. We also show that the cotangent of a double Lie groupoid is a symplectic double groupoid, and that the side groupoids of any symplectic double groupoid are Poisson groupoids in duality. Thus any double Lie groupoid gives rise to a dual pair of Poisson groupoids.