End (category theory)

In category theory, an end of a functor formula_1 is a universal extranatural transformation from an object "e" of X to "S".
More explicitly, this is a pair formula_2, where "e" is an object of X and formula_3 is an extranatural transformation such that for every extranatural transformation formula_4 there exists a unique morphism formula_5 
of X with formula_6 
for every object "a" of C.
By abuse of language the object "e" is often called the "end" of the functor "S" (forgetting formula_7) 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 formula_10 and the second is induced by formula_11.
Coend.
The definition of the coend of a functor formula_12 is the dual of the definition of an end.
Thus, a coend of "S" consists of a pair formula_13, where "d" is an object of X and formula_14
is an extranatural transformation, such that for every extranatural transformation formula_15 there exists a unique morphism
formula_16 of X with formula_17 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