End (category Theory) - Coend

The definition of the coend of a functor is the dual of the definition of an end.

Thus, a coend of S consists of a pair, where d is an object of X and

is a dinatural transformation, such that for every dinatural transformation

there exists a unique morphism

of X with

for every object a of C.

The coend d of the functor S is written

Dually, if X is cocomplete, then the coend can be described as the coequalizer in the diagram