Editing Initial and terminal objects (section)

=== Equivalent formulations ===

Terminal objects in a category {{mvar|C}} may also be defined as [[limit (category theory)|limit]]s of the unique empty [[diagram (category theory)|diagram]] {{math|'''0''' → ''C''}}. Since the empty category is vacuously a [[discrete category]], a terminal object can be thought of as an [[empty product]] (a product is indeed the limit of the discrete diagram {{math|{{mset|''X''<sub>''i''</sub>}}}}, in general). Dually, an initial object is a [[limit (category theory)|colimit]] of the empty diagram {{math|'''0''' → ''C''}} and can be thought of as an [[empty sum|empty]] [[coproduct]] or categorical sum.

It follows that any [[functor]] which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will take initial objects to initial objects. For example, the initial object in any [[concrete category]] with [[free object]]s will be the free object generated by the empty set (since the [[free functor]], being [[left adjoint]] to the [[forgetful functor]] to '''Set''', preserves colimits).

Initial and terminal objects may also be characterized in terms of [[universal property|universal properties]] and [[adjoint functors]]. Let '''1''' be the discrete category with a single object (denoted by •), and let {{math|''U'' : ''C'' → '''1'''}} be the unique (constant) functor to '''1'''. Then
* An initial object {{mvar|I}} in {{mvar|C}} is a [[universal morphism]] from • to {{mvar|U}}. The functor which sends • to {{mvar|I}} is left adjoint to ''U''.
* A terminal object {{mvar|T}} in {{mvar|C}} is a universal morphism from {{mvar|U}} to •. The functor which sends • to {{mvar|T}} is right adjoint to {{mvar|U}}.