  We give sufficient conditions to ensure that the ideal $\Phi(\mathcal E)$ of
$\mathcal E$-phantom maps in a locally $\lambda$-presentable exact category
$(\mathcal{A}, \mathcal{E})$ is (special) (pre)covering ideal, where $\mathcal
E$ is an exact substructure of $(\mathcal{A}, \mathcal{E})$. As a byproduct, we
infer the existence of various covering ideals in categories of sheaves which
have a meaningful geometrical motivation. In particular we deal with a
Zariski-local notion of phantom maps in categories of sheaves. We would like to
point out that our approach is necessarily different from [FGHT13], as the
categories involved in most of the examples we are interested in do not have
enough projective morphisms.
