  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. 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 $A$.
