Universal Property - Duality

Duality

Since the notions of initial and terminal are dual, it is often enough to discuss only one of them, and simply reverse arrows in C for the dual discussion. Alternatively, the word universal is often used in place of both words.

Note: some authors may call only one of these constructions a universal morphism and the other one a co-universal morphism. Which is which depends on the author, although in order to be consistent with the naming of limits and colimits the latter construction should be named universal and the former couniversal. This article uses the unambiguous terminology of initial and terminal objects.