End (category Theory)
In category theory, an end of a functor is a universal dinatural transformation from an object e of X to S.
More explicitly, this is a pair, where e is an object of X and
is a dinatural transformation from the constant functor whose value is e on every object and on every morphism, such that for every dinatural 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
If X is complete, the end can be described as the equalizer in the diagram
where the first morphism is induced by and the second morphism is induced by .