Editing Exact functor (section)

== Properties and theorems ==

A functor is exact if and only if it is both left exact and right exact.

A covariant (not necessarily additive) functor is left exact if and only if it turns finite [[limit (category theory)|limit]]s into limits; a covariant functor is right exact if and only if it turns finite [[colimit]]s into colimits; a contravariant functor is left exact iff it turns finite colimits into limits; a contravariant functor is right exact iff it turns finite limits into colimits. 

The degree to which a left exact functor fails to be exact can be measured with its [[derived functor|right derived functors]]; the degree to which a right exact functor fails to be exact can be measured with its [[derived functor|left derived functor]]s.

Left and right exact functors are ubiquitous mainly because of the following fact: if the functor ''F'' is [[adjoint functors|left adjoint]] to ''G'', then ''F'' is right exact and ''G'' is left exact.