Adding a constant and an axiom to a doctrine

Abstract: We study the meaning of "adding a constant to a language" for any doctrine, and "adding an axiom to a theory" for a primary doctrine, by showing how these are actually two instances of the same construction. We prove their universal properties, and how these constructions are compatible with additional structure on the doctrine. Existence of Kleisli object for comonads in the 2-category of indexed poset is proved in order to build these constructions.