Complete Lie algebroid actions and the integrability of Lie algebroids

We give a new proof of the equivalence between the existence of a complete action of a Lie algebroid on a surjective submersion and its integrability. The main tools in our approach are double Lie groupoids and multiplicative foliations, our proof relies on a simple characterization of those vacant double Lie groupoids which induce a Lie groupoid structure on their orbit spaces.
Comment: v2: minor corrections, v1:This note is a re-elaboration of Section 4 of arXiv:1910.06164v1