for every object of , there exists a unique morphism . The terminal object of any category, if it exists, is unique up to unique isomorphism. If the terminal object is also initial, it is called a zero object.
A terminal object is often written , since in Set it is a 1-element set, and also because it is the unit for the cartesian product. Other notations for a terminal object include and .
A terminal object may also be viewed as a limit over the empty diagram. Conversely, a limit over a diagram is a terminal cone over that diagram.
For any object in a category with terminal object , the categorical product and the exponential object both exist and are canonically isomorphic to .
Some examples of terminal objects in notable categories follow: