In category theory, an end of a functor is a universal extranatural transformation from an object e of X to S.^{[1]}
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
where the first morphism being equalized is induced by and the second is induced by .
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Transcription
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 an extranatural transformation, such that for every extranatural transformation there exists a unique morphism of X with for every object a of C.
The coend d of the functor S is written
Characterization as colimit: Dually, if X is cocomplete and C is small, then the coend can be described as the coequalizer in the diagram
Examples
 Natural transformations:
Suppose we have functors then
 .
In this case, the category of sets is complete, so we need only form the equalizer and in this case
the natural transformations from to . Intuitively, a natural transformation from to is a morphism from to for every in the category with compatibility conditions. Looking at the equalizer diagram defining the end makes the equivalence clear.
 Geometric realizations:
Let be a simplicial set. That is, is a functor . The discrete topology gives a functor , where is the category of topological spaces. Moreover, there is a map sending the object of to the standard simplex inside . Finally there is a functor that takes the product of two topological spaces.
Define to be the composition of this product functor with . The coend of is the geometric realization of .
Notes
References
 Mac Lane, Saunders (2013). Categories For the Working Mathematician. Springer Science & Business Media. pp. 222–226.
 Loregian, Fosco (2015). "This is the (co)end, my only (co)friend". arXiv:1501.02503 [math.CT].