The paper generalizes the concept of an affine set and an affine system. It is proved that the category of affine sets is isomorphic to a full coreflective subcategory of affine systems. A necessary and sufficient condition is given for the dual category of the variety of algebras whose objects underly the structure of affine sets to be isomorphic to a full reflective subcategory of affine systems. As a consequence, the sobriety-spatial equivalence for affine sets is obtained. A sufficient condition for the category of separated affine sets to make a reflective subcategory of the category of affine sets is shown. Illustrating examples are presented.