Editing Adjoint functors (section)

====Category theory====
* '''Equivalences.''' If ''F'' : ''D'' → ''C'' is an [[equivalence of categories]], then we have an inverse equivalence ''G'' : ''C'' → ''D'', and the two functors ''F'' and ''G'' form an adjoint pair. The unit and counit are natural isomorphisms in this case. If η : id → ''GF'' and ε : ''GF'' → id are natural isomorphisms, then there exist unique natural isomorphisms ε' : ''GF'' → id and η' : id → ''GF'' for which (η, ε') and (η', ε) are counit–unit pairs for ''F'' and ''G''; they are
*:<math>\varepsilon'=\varepsilon\circ(F\eta^{-1}G)\circ(FG\varepsilon^{-1})</math>
*:<math>\eta'=(GF\eta^{-1})\circ(G\varepsilon^{-1}F)\circ\eta</math>
* '''A series of adjunctions.''' The functor π<sub>0</sub> which assigns to a category its set of connected components is left-adjoint to the functor ''D'' which assigns to a set the discrete category on that set.  Moreover, ''D'' is left-adjoint to the object functor ''U'' which assigns to each category its set of objects, and finally ''U'' is left-adjoint to ''A'' which assigns to each set the indiscrete category<ref>{{cite web |title=Indiscrete category |url=http://ncatlab.org/nlab/show/indiscrete+category |website=nLab}}</ref> on that set.
* '''Exponential object'''. In a [[cartesian closed category]] the endofunctor ''C'' → ''C'' given by –×''A'' has a right adjoint –<sup>''A''</sup>. This pair is often referred to as [[currying]] and uncurrying; in many special cases, they are also continuous and form a homeomorphism.
<!--* '''Limits and Colimits.''' Limits and colimits can actually be viewed using adjoints when looking at functor categories. If C and D are two categories, then the functor '''limit''' from the category of functors from C to D to the category of constant functors from C to D which takes a given functor from C to D to its limit is in fact right-adjoint to the forgetful functor from the category of constant functors from C to D to the category of functors from C to D. Colimit is similarly the left-adjoint of this forgetful functor from the category of constant functors from C to D to the category of functors from C to D. -->