Zero Morphism does not Depend on Zero Object

Theorem
Let $\mathbf C$ be a category.

Let $A$ and $B$ be objects of $\mathbf C$.

Let $0_1$ and $0_2$ be zero objects of $\mathbf C$.

Then the morphism defined as the composition
 * $\beta \circ \alpha : A \to 0_1 \to B$

of the unique morphism $\alpha : A \to 0_1$ and the unique morphism $\beta : 0_1 \to B$ is equal to the morphism defined as the composition
 * $\delta \circ \gamma : A \to 0_2 \to B$

of the unique morphism $\gamma : A \to 0_2$ and the unique morphism $\delta : 0_2 \to B$.

Proof
There are unique morphisms $\epsilon : 0_1 \to 0_2$ and $\zeta : 0_2 \to 0_1$.

Since $0_1$ is terminal, we have
 * $\zeta \circ \epsilon = \operatorname{id}_{0_1}$
 * $\beta \circ \zeta = \delta$

Since $0_2$ is terminal, we have
 * $\epsilon \circ \alpha = \gamma$

Hence

Notation
This justifies the notation $0 := \beta \circ \alpha$, whenever a zero object exists in $\mathbf C$.