Localizations of varieties and quasivarieties (Q1568751)

It is well known that varieties (= monadic categories over sets) are nothing but exact categories having a regular projective regular generator with copowers. It is also well known that quasivarieties (= regular epireflective full subcategories of varieties) are nothing but regular categories having coequalizers of equivalence relations and a regular projective regular generator with copowers. \textit{E. M. Vitale} [J. Pure Appl. Algebra 108, No. 3, 315-320 (1996; Zbl 0854.18007)] succeeded in getting the characterization of localizations of varieties by dropping the projectivity assumption. The authors of this paper do the same thing to quasivarieties.