Higher-level constructs for families and multisets

In the earlier paper we proposed to use families instead of multisets to resolve the basic problems. In this paper we show that families instead of multisets support the higher-level constructions even, when no appropriate multiset-based constructions exist. We also show that multisets form a category and that the natural mapping from families to multisets is a functor. This emphasizes our claim that, for a definition of multisets, one should start with families and only introduce multiplicities as a secondary concept.