Editing Universal property (section)

===Limits and colimits===

Categorical products are a particular kind of [[limit (category theory)|limit]] in category theory. One can generalize the above example to arbitrary limits and colimits.

Let <math>\mathcal{J}</math> and <math>\mathcal{C}</math> be categories with <math>\mathcal{J}</math> a [[small category|small]] [[index category]] and let <math>\mathcal{C}^\mathcal{J}</math> be the corresponding [[functor category]]. The ''[[diagonal functor]]''
:<math>\Delta: \mathcal{C} \to \mathcal{C}^\mathcal{J}</math>
is the functor that maps each object <math>N</math> in <math>\mathcal{C}</math> to the constant functor <math>\Delta(N): \mathcal{J} \to \mathcal{C}</math> (i.e. <math>\Delta(N)(X) = N</math> for each <math>X</math> in <math>\mathcal{J}</math> and <math>\Delta(N)(f) = 1_N</math> for each <math>f: X \to Y</math> in <math>\mathcal{J}</math>) and each morphism <math>f : N \to M</math> in <math>\mathcal{C}</math> to the natural transformation <math>\Delta(f):\Delta(N)\to\Delta(M)</math> in <math>\mathcal{C}^{\mathcal{J}}</math> defined as, for every object <math>X</math> of <math>\mathcal{J}</math>, the component <math display="block">\Delta(f)(X):\Delta(N)(X)\to\Delta(M)(X) = f:N\to M</math>
at <math>X</math>. In other words, the natural transformation is the one defined by having constant component <math>f:N\to M</math> for every object of <math>\mathcal{J}</math>.

Given a functor <math>F: \mathcal{J} \to \mathcal{C}</math> (thought of as an object in <math>\mathcal{C}^\mathcal{J}</math>), the ''limit'' of <math>F</math>, if it exists, is nothing but a universal morphism from <math>\Delta</math> to <math>F</math>. Dually, the ''colimit'' of <math>F</math> is a universal morphism from <math>F</math> to <math>\Delta</math>.