Category with zero morphisms 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.
 A: 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 $X, Y, Z$. 
A: Zero morphism in a category $\mathcal C$ with Zero object $O$ from an object $A$ to $B$ which factors through zero object,that is the following diagram hold: 
$A\rightarrow O \to B$
Note that this factorization is unique as $O$ is both intial and terminal object.So zero morphism from an object $A$ to $B$ is unique.
A: Q: Does it mean that given $X,Y\in{\bf C}$, there can be only one morphism that we can use as $0_{XY}$?
Yes. Provided that the $0_{XY}$ form a complete collection, ie (i) given any (possibly equal) objects $X,Y$ in $\mathbf{C}$ there is a zero arrow $0_{XY}$, and (ii) these compose, ie $0_{YZ}\circ 0_{XY}=0_{XZ}$, then there are no other zero arrows.
Suppose that such a complete collection exists, and that for some objects $A,B$ the arrow $f_{AB}$ is a zero arrow. We have $0_{AB}=0_{BB}\circ 0_{AB}=0_{BB}\circ f_{AB}$ using (i) and (ii). But then $0_{BB}\circ f_{AB}=\text{Id}_B\circ f_{AB}=f_{AB}$ since $f_{AB}$ is a zero. Hence $f_{AB}=0_{AB}$. $\Box$
Note that one way of getting zero arrows is to find a zero object. But: (A) a category may have multiple zero objects. But different zero objects give the same zero arrows; and (B) we may have no zero objects, but still have a complete set of zero arrows. 
[Trivial example: one object and two arrows, the identity and the zero arrow. Less trivial: two objects, a complete set of zero arrows (4 of them), distinct identities (2 of them), and one other arrow (source one object, target the other).] 
