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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?

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    $\begingroup$ 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. $\endgroup$ – hardmath Jun 13 at 2:04
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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.

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