Zero morphism

Remarks

A category has at most one system of zero morphisms: if $0$ and $0'$ are two, then $0_{XY} = 0'_{YY} \circ 0_{XY} = 0'_{XY}$. If $\mathcal{C}$ has a zero object $0$, setting $0_{XY}$ equal to the composite of the unique morphisms $X \to 0$ and $0 \to Y$ gives a system of zero morphisms.

Example: in the category of groups, let $0_{GH}$ send every element of $G$ to the identity $e_H$ of $H$. For a group homomorphism $f \colon H \to K$, $f(e_H) = e_K$, so $f \circ 0_{GH} = 0_{GK}$; and $0_{GH} \circ g$ is again constant at $e_H$, so $0_{GH} \circ g = 0_{G'H}$ for $g \colon G' \to G$.

Non-example: the category of sets has no system of zero morphisms. There is no function at all from a non-empty set $X$ to the empty set, so $0_{X \emptyset}$ cannot be chosen.