Category with zero morphisms

Solution 1:

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 $X, Y, Z$.

Solution 2:

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.