Editing Limit (category theory) (section)

=== Universal property ===
Limits and colimits are important special cases of [[universal construction]]s.

Let ''C'' be a category and let ''J'' be a small index category. The [[functor category]] ''C''<sup>''J''</sup> may be thought of as the category of all diagrams of shape ''J'' in ''C''. The ''[[diagonal functor]]''
:<math>\Delta : \mathcal C \to \mathcal C^{\mathcal J}</math>
is the functor that maps each object ''N'' in ''C'' to the constant functor Δ(''N'') : ''J'' → ''C'' to ''N''. That is, Δ(''N'')(''X'') = ''N'' for each object ''X'' in ''J'' and Δ(''N'')(''f'') = id<sub>''N''</sub> for each morphism ''f'' in ''J''.

Given a diagram ''F'': ''J'' → ''C'' (thought of as an object in ''C''<sup>''J''</sup>), a [[natural transformation]] ''ψ'' : Δ(''N'') → ''F'' (which is just a morphism in the category ''C''<sup>''J''</sup>) is the same thing as a cone from ''N'' to ''F''. To see this, first note that Δ(''N'')(''X'') = ''N'' for all X implies that the components of ''ψ'' are morphisms ''ψ''<sub>''X''</sub> : ''N'' → ''F''(''X''), which all share the domain ''N''. Moreover, the requirement that the cone's diagrams commute is true simply because this ''ψ'' is a natural transformation. (Dually, a natural transformation ''ψ'' : ''F'' → Δ(''N'') is the same thing as a co-cone from ''F'' to ''N''.)

Therefore, the definitions of limits and colimits can then be restated in the form:
*A limit of ''F'' is a universal morphism from Δ to ''F''.
*A colimit of ''F'' is a universal morphism from ''F'' to Δ.