Editing Limit (category theory) (section)

=== As representations of functors ===
{{see also|Limit and colimit of presheaves}}
One can use [[Hom functor]]s to relate limits and colimits in a category ''C'' to limits in '''Set''', the [[category of sets]]. This follows, in part, from the fact the covariant Hom functor Hom(''N'', &ndash;) : ''C'' → '''Set''' [[#Preservation of limits|preserves all limits]] in ''C''. By duality, the contravariant Hom functor must take colimits to limits.

If a diagram ''F'' : ''J'' → ''C'' has a limit in ''C'', denoted by lim ''F'', there is a [[canonical isomorphism]]
:<math>\operatorname{Hom}(N,\lim F)\cong \lim\operatorname{Hom}(N,F-)</math>
which is natural in the variable ''N''. Here the functor Hom(''N'', ''F''&ndash;) is the composition of the Hom functor Hom(''N'', &ndash;) with ''F''. This isomorphism is the unique one which respects the limiting cones.

One can use the above relationship to define the limit of ''F'' in ''C''. The first step is to observe that the limit of the functor Hom(''N'', ''F''&ndash;) can be identified with the set of all cones from ''N'' to ''F'':
:<math>\lim\operatorname{Hom}(N,F-) = \operatorname{Cone}(N,F).</math>
The limiting cone is given by the family of maps π<sub>''X''</sub> : Cone(''N'', ''F'') → Hom(''N'', ''FX'') where {{pi}}<sub>''X''</sub>(''ψ'') = ''ψ''<sub>''X''</sub>. If one is given an object ''L'' of ''C'' together with a [[natural isomorphism]] ''Φ'' : Hom(''L'', &ndash;) → Cone(&ndash;, ''F''), the object ''L'' will be a limit of ''F'' with the limiting cone given by ''Φ''<sub>''L''</sub>(id<sub>''L''</sub>). In fancy language, this amounts to saying that a limit of ''F'' is a [[representable functor|representation]] of the functor Cone(&ndash;, ''F'') : ''C'' → '''Set'''.

Dually, if a diagram ''F'' : ''J'' → ''C'' has a colimit in ''C'', denoted colim ''F'', there is a unique canonical isomorphism
:<math>\operatorname{Hom}(\operatorname{colim} F, N)\cong\lim\operatorname{Hom}(F-,N)</math>
which is natural in the variable ''N'' and respects the colimiting cones. Identifying the limit of Hom(''F''&ndash;, ''N'') with the set Cocone(''F'', ''N''), this relationship can be used to define the colimit of the diagram ''F'' as a representation of the functor Cocone(''F'', &ndash;).