  A quasivariety has the weak ES property when the epimorphisms between its
finitely generated members are surjective. A characterization of quasivarieties
with the weak ES property is obtained and a method for detecting failures of
this property in quasivarieties with a near unanimity term and in congruence
permutable varieties is given. It is also shown that under reasonable
assumptions the weak ES property implies arithmeticity. In particular, every
filtral variety with the weak ES property is a discriminator variety.
