Editing Adjoint functors (section)

===Diagonal functors and limits===
[[Product (category theory)|Products]], [[Pullback (category theory)|fibred products]], [[Equalizer (mathematics)|equalizers]], and [[Kernel (algebra)|kernels]] are all examples of the categorical notion of a [[limit (category theory)|limit]].  Any limit functor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question), and the counit of the adjunction provides the defining maps from the limit object (i.e. from the diagonal functor on the limit, in the functor category).  Below are some specific examples.

* '''Products''' Let Π : '''Grp<sup>2</sup>''' → '''Grp''' be the functor that assigns to each pair (''X''<sub>1</sub>, ''X<sub>2</sub>'') the product group ''X''<sub>1</sub>×''X''<sub>2</sub>, and let Δ : '''Grp →''' '''Grp<sup>2</sup>'''  be the [[diagonal functor]] that assigns to every group ''X'' the pair (''X'', ''X'') in the product category '''Grp<sup>2</sup>'''. The universal property of the product group shows that Π is right-adjoint to Δ.  The counit of this adjunction is the defining pair of projection maps from ''X''<sub>1</sub>×''X''<sub>2</sub> to ''X''<sub>1</sub> and ''X''<sub>2</sub> which define the limit, and the unit is the ''diagonal inclusion'' of a group X into ''X''×''X'' (mapping x to (x,x)).

: The [[cartesian product]] of [[Set (mathematics)|sets]], the product of rings, the [[product topology|product of topological spaces]] etc. follow the same pattern; it can also be extended in a straightforward manner to more than just two factors.  More generally, any type of limit is right adjoint to a diagonal functor.

* '''Kernels.''' Consider the category ''D'' of homomorphisms of abelian groups. If ''f''<sub>1</sub> : ''A''<sub>1</sub> → ''B''<sub>1</sub> and ''f''<sub>2</sub> : ''A''<sub>2</sub> → ''B''<sub>2</sub> are two objects of ''D'', then a morphism from ''f''<sub>1</sub> to ''f''<sub>2</sub> is a pair (''g''<sub>''A''</sub>, ''g''<sub>''B''</sub>) of morphisms such that ''g''<sub>''B''</sub>''f''<sub>1</sub> = ''f''<sub>2</sub>''g''<sub>''A''</sub>. Let ''G'' : ''D'' → '''Ab''' be the functor which assigns to each homomorphism its [[kernel (algebra)|kernel]] and let ''F'' : '''Ab →''' ''D''  be the functor which maps the group ''A'' to the homomorphism ''A'' → 0. Then ''G'' is right adjoint to ''F'', which expresses the universal property of kernels.  The counit of this adjunction is the defining embedding of a homomorphism's kernel into the homomorphism's domain, and the unit is the morphism identifying a group ''A'' with the kernel of the homomorphism ''A'' → 0.

: A suitable variation of this example also shows that the kernel functors for vector spaces and for modules are right adjoints. Analogously, one can show that the cokernel functors for abelian groups, vector spaces and modules are left adjoints.