Equivalence of Categories

A pair of functors F : C → D and G : D → C together with natural isomorphisms ε : F∘G ⇒ Id_D and η : Id_C ⇒ G∘F (or equivalently F fully faithful and essentially surjective) showing that C and D have the same categorical structure up to isomorphism of objects; not necessarily a strict isomorphism of categories but a weaker notion preserving categorical properties.

A relationship between categories C and D given by functors F : C → D and G : D → C together with natural isomorphisms η: Id_C ⇒ G∘F and ε: F∘G ⇒ Id_D; equivalently F is essentially surjective on objects and fully faithful, so C and D have the same categorical structure up to coherent isomorphism.

A functor F: C → D is an equivalence of categories if it is fully faithful (induces bijections on hom-sets) and essentially surjective (every object of D is isomorphic to F(c) for some c in C); equivalence identifies categories up to isomorphism of objects rather than strict equality.