Let F, G:ℭ->D be functors between categories ℭ and D. A natural transformation Φ from F to G consists of a family Φ_C :F(C)->G(C) of morphisms in D which are indexed by the objects C of ℭ so that, for each morphism f:C->D between objects in ℭ, the equality G(f)°Φ_C = Φ_D °F(f):F(C)->G(D) holds. The elements Φ_C are called the components of the natural transformation. If all the components Φ_C are isomorphisms in D, then Φ is called a natural isomorphism between F and G. In this case, one writes Φ:F≃G.

category | category theory | functor | morphism | natural isomorphism | object | tensor category | unital natural transformation