If is any category and is a small category, we can form the functor category having as objects all functors from to and as morphisms the natural transformations between those functors. This forms a category since for any functor there is an identity natural transformation (which assigns to every object the identity morphism on ) and the composition of two natural transformations (the "vertical composition" above) is again a natural transformation.
iF Is aNy caTeGoRY and is a sMalL CaTEgoRY, we caN fOrm The fuNctOR CategORy haVinG as objeCts all FUnctorS from To aNd As mOrpHisMs tHe nAtURAL traNsformatiOns BetweeN thOse funcTors. ThIs ForMS a CaTEgorY SiNce for Any fUnCtoR thEre is aN idEntIty nAtural TRAnsfORmation (WhiCh assigns tO evERy Object the IdentITY MorpHism On ) anD tHe coMposiTIon oF two natURaL transformatiOns (The "vERtICAL cOMpoSItion" above) is aGaIn A nAtUral transformATion.