Editing Limit (category theory) (section)

===Colimits===
{{see also|Direct limit}}
The [[Dual (category theory)|dual notions]] of limits and cones are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here:

A '''[[co-cone]]''' of a diagram <math>F:J\to C</math> is an object <math>N</math> of <math>C</math> together with a family of morphisms
:<math>\psi_X:F(X) \to N</math>

for every object <math>X</math> of <math>J</math>, such that for every morphism <math>f:X\to Y</math> in <math>J</math>, we have <math>\psi_Y\circ F(f)=\psi_X</math>.

A '''colimit''' of a diagram <math>F:J\to C</math> is a co-cone <math>(L, \phi)</math> of <math>F</math> such that for any other co-cone <math>(N, \psi)</math> of <math>F</math> there exists a unique morphism <math>u:L\to N</math> such that <math>u\circ \phi_X = \psi_X</math> for all <math>X</math> in <math>J</math>.

[[File:Functor co-cone (extended).svg|center|A universal co-cone]]

Colimits are also referred to as ''[[universal co-cone]]s''. They can be characterized as [[initial object]]s in the [[category of co-cones]] from <math>F</math>.

As with limits, if a diagram <math>F</math> has a colimit then this colimit is unique up to a unique isomorphism.