An initial object in a category is an object such that for any object of , there is a unique morphism . An initial object, if it exists, is unique up to unique isomorphism, so we speak of the initial object.
An initial object may also be called coterminal, universal initial, co-universal, or simply universal.
Initial objects are the dual concept to terminal objects: an initial object in is the same as a terminal object in . An object that is both initial and terminal is called a zero object.