Abstract: This chapter focuses on abstract characterizations. It is shown that one of the basic ways of giving abstract characterization is via a Universal Mapping Property (UMP). Epis and monos, initial and terminal objects, generalized elements, sections and retractions, products, categories with products, and hom-sets are discussed. Exercises are provided in the last part of the chapter.

Keywords: abstract characterization, Universal Mapping Property, epis and monos, initial objects, terminal objects, generalized elements, hom-sets,