End (category theory)

In category theory, an end of a functor ${\displaystyle S:\mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X} }$ is a universal extranatural transformation from an object e of X to S.

More explicitly, this is a pair ${\displaystyle (e,\omega )}$, 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 ${\displaystyle \omega }$) and is written

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