Editing Initial and terminal objects (section)

=== Existence and uniqueness ===

Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if {{math|''I''<sub>1</sub>}} and {{math|''I''<sub>2</sub>}} are two different initial objects, then there is a unique [[isomorphism]] between them. Moreover, if {{mvar|I}} is an initial object then any object isomorphic to {{mvar|I}} is also an initial object. The same is true for terminal objects.

For [[complete category|complete categories]] there is an existence theorem for initial objects. Specifically, a ([[locally small category|locally small]]) complete category {{mvar|C}} has an initial object if and only if there exist a set {{mvar|I}} ({{em|not}} a [[proper class]]) and an {{mvar|I}}-[[indexed family]] {{math|(''K''<sub>''i''</sub>)}} of objects of {{mvar|C}} such that for any object {{mvar|X}} of {{mvar|C}}, there is at least one morphism {{math|''K''<sub>''i''</sub> → ''X''}} for some {{math|''i'' ∈ ''I''}}.