We characterize localizations of monadic categories over SET using the fact that the category of algebras for a monad over SET is the exact completion of the full subcategory of free algebras. This also constitutes an unifying argument to characterize reflections and epireflections.
Vitale, E. (1996). Localizations of algebraic categories. Journal of Pure and Applied Algebra, 108(3), 315-320. https://doi.org/10.1016/0022-4049(95)00076-3 (Original work published 1996)