stylistic:tone
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 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

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 form the components of a natural transformation

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 functors. For contravariant functors the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.