Hi all:

I was reading a proof that localization of categories preserves additivity. In the proof the author uses a statement:

If $X$ is an object in a preadditive category $C$ and ${End}_C(X) = \{0_X\}$ then $X$ is the zero object in $C$.

I know that the statement is true when $C$ is an additive category, but I'm not quite sure if it's true for preadditive category.

To be precise, I traced back to the very definition of a zero object, that is, both initial and terminal. However, the condition in the statement only suggests the uniqueness of morphism $X\rightarrow Y$ for any object $Y\in \text{ob}C$, I don't see why such a morphism should exist.

Any suggestions or reference? Thanks.