This paper introduces a generalization of the concept of Set category introduced in [10] by constructing the category - whose objects are small ℒ - fuzzy sets in which the characteristic functions takes its values from a complete distributive lattice, and its arrows are ℒ - fuzzy maps. After that we construct a functor - between these two categories, in a way that forgets the fuzziness of sets and maps, and formalizing the inclusion functor - . In addition, we study of the applications of universal arrows in category - , and getting back to the classical state and comparing it with that introduced in [10].