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Let ${\bf C}$ be a category with zero morphisms, i.e., for each $X,Y\in {\bf C}$, there is a morphism $0_{XY}:X\rightarrow Y$ satisfying certain properties ($0_{XY}$ composes with every morphism from $Y$ or to $X$ to give another such). According to Wikipedia, "the collection of $0_{XY}$ is unique." Does it mean that given $X,Y\in{\bf C}$, there can be only one morphism that we can use as $0_{XY}$? When defining kernel and cokernel, one uses these morphisms $0_{XY}$, so it appears as though kernel and cokernel depends on a particular choice of the family of $0_{XY}$, which is very unpleasant.

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Yes. The idea is the same as the proof of the uniqueness of identities in a monoid. If $f_{X, Y} : X \to Y$ is a family of zero morphisms and $g_{X, Y} : X \to Y$ is another family of zero morphisms, then

$$f_{Y, Z} \circ g_{X, Y} = g_{X, Z} = f_{X, Z}$$

for every triple of objects $X, Y, Z$.

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