Limit (category theory)
Adjunctions

Like all universal constructions, the formation of limits and colimits is functorial in nature. In other words, if every diagram of shape J has a limit in C (for J small) there exists a limit functor

which assigns each diagram its limit and each natural transformation η : F → G the unique morphism lim η : lim F → lim G commuting with the corresponding universal cones. This functor is right adjoint to the diagonal functor Δ : C → C.This adjunction gives a bijection between the set of all morphisms from N to lim F and the set of all cones from N to F

which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected (and nonempty) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ. This fails if J is not connected. For example, if J is a discrete category, the components of the unit are the diagonal morphisms δ : N → N.

Dually, if every diagram of shape J has a colimit in C (for J small) there exists a colimit functor

which assigns each diagram its colimit. This functor is left adjoint to the diagonal functor Δ : C → C, and one has a natural isomorphism

The unit of this adjunction is the universal cocone from F to colim F. If J is connected (and nonempty) then the counit is an isomorphism, so that colim is a left inverse of Δ.

Note that both the limit and the colimit functors are covariant functors.
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