On categories of algebras equivalent to a quasivariety

In a recent paper, B. Banaschewski proved that anySP-class of algebras which is category equivalent to a variety (over a possibly different finitary similarity type) is itself a variety. Here we prove the analogous statement obtained by replacing “variety” with “quasivariety”. We also present examples which detail some of the difficulties arising when one tries to strengthen the theorem in various ways.