for every object $x$ of $C$, there exists a unique morphism$!:x\to 1$. 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.

Remarks

A terminal object is often written $1$, 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 $pt$.

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 $x$ in a category with terminal object $1$, the categorical product$x\times 1$ and the exponential object$x^1$ both exist and are canonically isomorphic to $x$.

Examples

Some examples of terminal objects in notable categories follow:

The terminal object of a poset is its top element, if it exists.