A category C together with a covariant functor T from C to the category Top of all topological spaces allows us to construct a category C T₀ of pairs (M, A) as objects, where A is a set of points of the set of all points of T(M) . Next, a covariant functor L from a subcategory of C T₀ to C is considered. In a category C equipped with covariant functors T and L satisfying some natural axioms of localization the concepts of regularity, weak regularity, coregularity and weak coregularity of morphisms of C is introduced and some categorial features of these concepts are established.