VBVB-algebroid morphisms and representations up to homotopy

We show in this paper that the correspondence between 2-term representations up to homotopy and VBVB-algebroids, established in [6], holds also at the level of morphisms. This correspondence is hence an equivalence of categories. As an application, we study foliations and distributions on a Lie algebroid, that are compatible both with the linear structure and the Lie algebroid structure. In particular, we show how infinitesimal ideal systems in a Lie algebroid A are related with subrepresentations of the adjoint representation of A.