In this paper we continue with the study of the category MAP of continuous maps and their morphisms, introduced in [3]. This category is an extension of both the category TOP_Y (of continuous maps into a fixed space Y and their morphisms) and TOP (of topological spaces and continuous maps as morphisms). Partial products are used to obtain universal type theorems for T_0, Tychonoff and zero-dimensional maps. Finally we introduce zero-dimensional and strongly zero-dimensional maps and generalize some well known results in the category TOP concerning zero-dimensional and strongly zero-dimensional spaces to the category MAP.