  We prove that, under suitable assumptions on a category C, the existence of
supercompact cardinals implies that every absolute epireflective class of
objects of C is a small-orthogonality class. More precisely, if L is a
localization functor on an accessible category C such that the unit morphism X
\to LX is an extremal epimorphism for all X, and the class of L-local objects
is defined by an absolute formula with parameters, then the existence of a
supercompact cardinal above the cardinalities of the parameters implies that L
is a localization with respect to some set of morphisms.
