Zero morphism does not depend on zero object

Question

I want to prove that zero morphism does not depend on zero object. Let a zero object $0 \in \mathcal{C}$. I supposed that exists another zero object $0' \in \mathcal{C}$, and I considered $0: A \to B$ and also $$A \to 0 \to 0' \to B$$ But I don't know to prove that zero morphism $0: A \to B$ not does not depend on zero object. Can anybody give me a suggestion?

Answer 1

A zero object in a category is an object, which is both initial and terminal. As such it is unique up to unique isomorphism.

A zero morphism in a category with zero object is a morphism which factors over a zero object.

Let $0:X \rightarrow 0 \rightarrow Y$ and $0‘:X \rightarrow 0‘ \rightarrow Y$ be two zero morphisms from $X$ to $Y$. Then $0 \rightarrow Y = 0 \rightarrow 0‘ \rightarrow Y$ and $X \rightarrow 0‘ = X \rightarrow 0 \rightarrow 0‘$ by the universal properties of the zero objects. This shows that both zero morphisms $0=X \rightarrow 0 \rightarrow 0‘ \rightarrow Y=0‘$.

As far as I know you can also define a category with zero morphisms without having zero objects. For example you can take monoid-enriched categories. Zero objects yield such an enrichment.