Editing Limit (category theory) (section)

=== 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'''
:<math>\lim : \mathcal{C}^\mathcal{J} \to \mathcal{C}</math>
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''<sup>''J''</sup>.
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''
:<math>\operatorname{Hom}(N,\lim F) \cong \operatorname{Cone}(N,F)</math>
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 category|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 morphism]]s δ : ''N'' → ''N''<sup>''J''</sup>.

Dually, if every diagram of shape ''J'' has a colimit in ''C'' (for ''J'' small) there exists a '''colimit functor'''
:<math>\operatorname{colim} : \mathcal{C}^\mathcal{J} \to \mathcal{C}</math>
which assigns each diagram its colimit. This functor is [[left adjoint]] to the diagonal functor Δ : ''C'' → ''C''<sup>''J''</sup>, and one has a natural isomorphism
:<math>\operatorname{Hom}(\operatorname{colim}F,N) \cong \operatorname{Cocone}(F,N).</math>
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 functor|''covariant'']] functors.