Definition
Formally, given two categories C and D, an equivalence of categories consists of a functor F : C → D, a functor G : D → C, and two natural isomorphisms ε: FG→ID and η : IC→GF. Here FG: D→D and GF: C→C, denote the respective compositions of F and G, and IC: C→C and ID: 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 if an inverse functor G and natural isomorphisms as above exist. Note however that knowledge of F is usually not enough to reconstruct G and the natural isomorphisms: there may be many choices (see example below).
Read more about this topic: Equivalence Of Categories
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