Initial and terminal objects

In category theory, a branch of mathematics, an initial object of a category $C$ is an object $I$ in $C$ such that for every object $X$ in $C$ , there exists precisely one morphism $I \to X$ .

The dual notion is that of a terminal object (also called terminal element): $T$ is terminal if for every object $X$ in $C$ there exists exactly one morphism $X \to T$ . Initial objects are also called coterminal or universal, and terminal objects are also called final.

If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object.

A strict initial object $I$ is one for which every morphism into $I$ is an isomorphism (strict terminal objects are defined analogously).