  In this note a functorial approach to the integration problem of an
LA-groupoid to a double Lie groupoid is discussed. To do that, we study the
notions of fibred products in the categories of Lie groupoids and Lie
algebroids, giving necessary and sufficient conditions for the existence of
such. In particular, it turns out, that the fibred product of Lie algebroids
along integrable morphisms is always integrable by a fibred product of Lie
groupoids. We show that to every LA-groupoid with integrable top structure one
can associate a differentiable graph in the category of Lie groupoids, which is
an integrating double Lie groupoid, whenever some lifting conditions for
suitable Lie algebroid homotopies are fulfilled; the result specializes to the
case of a Poisson groupoid, yielding a symplectic double groupoid, provided our
conditions on the associated LA-groupoid are satisfied.
