Initial Object

In a category $\mathcal{C}$, $A$ is an initial object iff

$$\forall B:\mathcal{C}.\exists !f:A\to B.$$

Remarks

All initial objects in $\mathcal{C}$ are terminal objects in ${\mathcal{C}}^{op}$. All initial objects are isomorphic. If an terminal object is also an initial object then it is called a zero object or a “null object”, and the category is considered pointed. An initial object is an example of a colimit of the empty diagram.