Zero object

Remarks

A zero object is unique up to unique isomorphism, because an initial object is. A category with a zero object is called pointed.

Example: in the category of groups, the trivial group $\{e\}$ is a zero object. The only homomorphism $\{e\} \to G$ sends $e$ to the identity of $G$, and the only map $G \to \{e\}$ is the constant map, which is a homomorphism. The same check shows that the zero module is a zero object in the category of modules over a ring. It also shows that a one-point set is a zero object in the category of pointed sets.

Non-example: in the category of sets, a one-point set $\{*\}$ is terminal but not initial, since there are two functions $\{*\} \to \{0, 1\}$. The empty set is initial but not terminal, since there is no function from a non-empty set to it. So the category of sets has no zero object.