# Zero morphism does not depend on zero object

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?

• I surmise from the tag "category-theory" that this is the context of your Question. But you omit specific mention of what kind of category is assumed (it appears only as a tag). Probably the setup of your Question needs to articulate how "zero morphism" and/or "zero object" are to be defined. – hardmath Jun 13 at 2:04

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 $$0=X \rightarrow 0 \rightarrow 0‘ \rightarrow Y=0‘$$.