This is the second of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of L-topological spaces. In the first paper in this series, subtitled Basic Concepts, the authors presented the notions of direlation and difunction between textures and introduced the category dfTex, the construct fTex and several related categories. In this paper the category dfDitop of ditopological texture spaces and bicontinuous difunctions is defined and the forgetful functor U: dfDitop --> dfTrex shown to be topological. Several properties are discussed, including the existence of products and coproducts. An equivalence with the category of classical Hutton spaces is presented and the paper ends with a consideration of L-valued sets and topologies from the viewpoint of ditopological texture spaces. (C) 2004 Elsevier B.V. All rights reserved.