zero object

An initial object in a category $\mathcal{C}$ is an object $A$ in $\mathcal{C}$ such that, for every object $X$ in $\mathcal{C}$ , there is exactly one morphism $A \longrightarrow X$ .

A terminal object in a category $\mathcal{C}$ is an object $B$ in $\mathcal{C}$ such that, for every object $X$ in $\mathcal{C}$ , there is exactly one morphism $X \longrightarrow B$ .

A zero object in a category $\mathcal{C}$ is an object $0$ that is both an initial object and a terminal object.

All initial objects (respectively, terminal objects, and zero objects), if they exist, are isomorphic in $\mathcal{C}$ .