Editing Initial and terminal objects (section)

=== Relation to other categorical constructions ===

Many natural constructions in category theory can be formulated in terms of finding an initial or terminal object in a suitable category.

* A [[universal morphism]] from an object {{mvar|X}} to a functor {{mvar|U}} can be defined as an initial object in the [[comma category]] {{math|(''X'' ↓ ''U'')}}. Dually, a universal morphism from {{mvar|U}} to {{mvar|X}} is a terminal object in {{math|(''U'' ↓ ''X'')}}.
* The limit of a diagram {{mvar|F}} is a terminal object in {{math|Cone(''F'')}}, the [[category of cones]] to {{mvar|F}}. Dually, a colimit of {{mvar|F}} is an initial object in the category of cones from {{mvar|F}}.
* A [[representable functor|representation of a functor]] {{mvar|F}} to '''Set''' is an initial object in the [[category of elements]] of {{mvar|F}}.
* The notion of [[final functor]] (respectively, initial functor) is a generalization of the notion of final object (respectively, initial object).