In category theory, an end of a functor S : C o p × C → X {\displaystyle S:\mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X} } is a universal extranatural transformation from an object e of X to S.More explicitly, this is a pair {\displaystyle }, where e is an object of X and ω : e → ¨ S {\displaystyle \omega :e{\ddot {\to }}S} is an extranatural transformation such that for every extranatural transformation β : x → ¨ S {\displaystyle \beta :x{\ddot {\to }}S} there exists a unique morphism h : x → e {\displaystyle h:x\to e} of X with β a = ω a ∘ h {\displaystyle \beta _{a}=\omega _{a}\circ h} for every object a of C. By abuse of language the object e is often called the end of the functor S and is written e = ∫ c S or just ∫ C S.
