This paper is the third paper of a series devoted to higher-dimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition systems. In this sequel, it is proved that there exists a model category of labelled symmetric precubical sets which is Quillen equivalent to the Bousfield localization of this left determined model category by the cubification functor. The realization functor from labelled symmetric precubical sets to cubical transition systems which was introduced in the first paper of this series is used to establish this Quillen equivalence. However, it is not a left Quillen functor. It is only a left adjoint. It is proved that the two model categories are related to each other by a zig-zag of Quillen equivalences of length two. The middle model category is still the model category of cubical transition systems, but with an additional family of generating cofibrations. The weak equivalences are closely related to bisimulation. Similar results are obtained by restricting the constructions to the labelled symmetric precubical sets satisfying the HDA paradigm.