Editing Limit (category theory) (section)

===Preservation of limits===
A functor ''G'' : ''C'' → ''D'' induces a map from Cone(''F'') to Cone(''GF''): if ''Ψ'' is a cone from ''N'' to ''F'' then ''GΨ'' is a cone from ''GN'' to ''GF''. The functor ''G'' is said to '''preserve the limits of ''F''''' if (''GL'', ''Gφ'') is a limit of ''GF'' whenever (''L'', ''φ'') is a limit of ''F''. (Note that if the limit of ''F'' does not exist, then ''G'' [[vacuous truth|vacuously]] preserves the limits of ''F''.)

A functor ''G'' is said to '''preserve all limits of shape ''J''''' if it preserves the limits of all diagrams ''F'' : ''J'' → ''C''. For example, one can say that ''G'' preserves products, equalizers, pullbacks, etc. A '''continuous functor''' is one that preserves all ''small'' limits.

One can make analogous definitions for colimits. For instance, a functor ''G'' preserves the colimits of ''F'' if ''G''(''L'', ''φ'') is a colimit of ''GF'' whenever (''L'', ''φ'') is a colimit of ''F''. A '''cocontinuous functor''' is one that preserves all ''small'' colimits.

If ''C'' is a [[complete category]], then, by the above existence theorem for limits, a functor ''G'' : ''C'' → ''D'' is continuous if and only if it preserves (small) products and equalizers. Dually, ''G'' is cocontinuous if and only if it preserves (small) coproducts and coequalizers.

An important property of [[adjoint functors]] is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous. Since adjoint functors exist in abundance, this gives numerous examples of continuous and cocontinuous functors.

For a given diagram ''F'' : ''J'' → ''C'' and functor ''G'' : ''C'' → ''D'', if both ''F'' and ''GF'' have specified limits there is a unique canonical morphism
:<math>\tau_F : G \lim F \to \lim GF</math>
which respects the corresponding limit cones. The functor ''G'' preserves the limits of ''F'' if and only if this map is an isomorphism. If the categories ''C'' and ''D'' have all limits of shape ''J'' then lim is a functor and the morphisms τ<sub>''F''</sub> form the components of a [[natural transformation]]
:<math>\tau:G \lim \to \lim G^J.</math>
The functor ''G'' preserves all limits of shape ''J'' if and only if τ is a natural isomorphism. In this sense, the functor ''G'' can be said to ''commute with limits'' ([[up to]] a canonical natural isomorphism).

Preservation of limits and colimits is a concept that only applies to ''[[covariant functor|covariant]]'' functors. For [[contravariant functor]]s the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.