Editing Adjoint functors (section)

===Colimits and diagonal functors===
[[Coproduct]]s, [[Pushout (category theory)|fibred coproducts]], [[coequalizer]]s, and [[cokernel]]s are all examples of the categorical notion of a [[limit (category theory)|colimit]].  Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object.  Below are some specific examples.

* '''Coproducts.''' If ''F'' :  '''Ab'''<sup>'''2'''</sup> '''→''' '''Ab''' assigns to every pair (''X''<sub>1</sub>, ''X''<sub>2</sub>) of abelian groups their [[Direct sum of groups|direct sum]], and if ''G'' : '''Ab''' → '''Ab'''<sup>'''2'''</sup> is the functor which assigns to every abelian group ''Y'' the pair (''Y'', ''Y''), then ''F'' is left adjoint to ''G'', again a consequence of the universal property of direct sums.  The unit of this adjoint pair is the defining pair of inclusion maps from ''X''<sub>1</sub> and ''X''<sub>2</sub> into the direct sum, and the counit is the additive map from the direct sum of (''X'',''X'') to back to ''X'' (sending an element (''a'',''b'') of the direct sum to the element ''a''+''b'' of ''X'').

: Analogous examples are given by the [[Direct sum of modules|direct sum]] of [[vector space]]s and [[module (mathematics)|modules]], by the [[free product]] of groups and by the disjoint union of sets.