VB-algebroid morphisms and representations up to homotopy

We show in this paper that the correspondence between 2-term representations up to homotopy and $\mathcal{VB}$-algebroids, established by Gracia-Saz and Mehta, holds also at the level of morphisms, proving in particular that such correspondence is an equivalence of categories. As an application we study the relationship between IM-foliations on a Lie algebroid $A$ (the infinitesimal notion of ideal systems) and subrepresentations of the adjoint representation of $A$.