In category theory, an end of a functor is a universal extranatural transformation from an object e of X to S. More explicitly, this is a pair , where e is an object of X and is an extranatural transformation such that for every extranatural transformation there exists a unique morphism of X with for every object a of C. By abuse of language the object e is often called the end of the functor S (forgetting ) and is written Characterization as limit: If X is complete and C is small, the end can be described as the equalizer in the diagram