This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions.
Because of the faithfulness of the functor U, the homomorphisms of a concrete category may be formally identified with their underlying functions (i.e., their images under U); the homomorphisms then regain the usual interpretation as "structure-preserving" functions.
(Item 6 under Further examples expresses the same U in less elementary language via presheaves.)
The requirement that U be faithful means that it maps different morphisms between the same objects to different functions.
Similarly, any set with four elements can be given two non-isomorphic group structures: one isomorphic to
The fact that there does not exist any faithful functor from hTop to Set was first proven by Peter Freyd.
Given a concrete category (C, U) and a cardinal number N, let UN be the functor C → Set determined by UN(c) = (U(c))N. Then a subfunctor of UN is called an N-ary predicate and a natural transformation UN → U an N-ary operation.
For example, it may be useful to think of the models of a theory with N sorts as forming a concrete category over SetN.