respective compositions of F and G, and I_C: C->C and I_D: D->D denote the identity functors on C and D, assigning each object and morphism to itself. If F and G are contravariant functors one speaks of a duality of categories instead.

One often does not specify all the above data. For instance, we say that the categories C and D are equivalent (respectively dually equivalent) if there exists an equivalence (respectively duality) between them. Furthermore, we say that F "is" an equivalence of categories