Editing Adjoint functors (section)

===Limit preservation===

The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore ''is'' a right adjoint) is ''continuous'' (i.e. commutes with [[limit (category theory)|limits]] in the category theoretical sense); every functor that has a right adjoint (and therefore ''is'' a left adjoint) is ''cocontinuous'' (i.e. commutes with [[limit (category theory)|colimits]]).

Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. For example:
* applying a right adjoint functor to a [[product (category theory)|product]] of objects yields the product of the images;
* applying a left adjoint functor to a [[coproduct]] of objects yields the coproduct of the images;
* every right adjoint functor between two abelian categories is [[left exact functor|left exact]];
* every left adjoint functor between two abelian categories is [[right exact functor|right exact]].