Definition:Zero Morphism via Zero Object

Definition
Let $C$ be a category with has a zero object $0$.

Let $a,b\in C$ be objects.

The zero morphism $0$ from $a$ to $b$ is the composition of the unique morphism $a \to 0$ and the unique morphism $0 \to b$:
 * $0 : a \to 0 \to b$

Also see

 * Zero Morphism does not Depend on Zero Object