Exactness of cochain complexes via additive functors

We investigate the relation between the notion of $e$-exactness, recently introduced by Akray and Zebary, and some functors naturally related to it, such as the functor $P\text{\hspace{0.17em}⁣}:\mathrm{Mod}R\to \mathrm{Spec}(\mathrm{Mod}R)$, where $\mathrm{Spec}(\mathrm{Mod}R)$ denotes the spectral category of $\mathrm{Mod}R$, and the localization functor with respect to the singular torsion theory.