Abstract: We introduce a model structure on the category of cubical sets which is constructed using our model structure on marked cubical sets from part 1, and whose fibrant objects are defined by the right lifting property with respect to open boxes with certain edges degenerate. We discuss the proof that this model structure is Quillen-equivalent to the Joyal model structure on simplicial sets. We will also introduce the homotopy category and mapping spaces of a fibrant cubical set, and characterize weak equivalences between fibrant cubical sets in terms of these concepts.