nLab
zero morphism

In a category $C$ with zero object $0$ the zero morphism $0_{c,d}:c\to d$ between two objects $c,d\in C$ is the unique morphism that factors through $0$:

$$0_{c,d}:c\to 0\to d.$$0_{c,d} : c \to 0 \to d \,.

See zero object for examples.

More generally, in any category enriched over the closed monoidal category of pointed sets (with smash product), the zero morphism $0_{c,d}:c\to d$ is the basepoint of the hom-object $[c,d]$.

In fact, an enrichment over pointed sets consists precisely of the choice of a ‘zero’ morphism $0_{c,d}:c\to d$ for each pair of objects, with the property that $0_{c,d}\circ f=0_{b,d}$ and $f\circ 0_{a,b}=0_{a,c}$ for any morphism $f:b\to c$. Such an enrichment is unique if it exists, for if we are given a different collection of zero morphisms $0{\prime }_{c,d}$, we must have

$$0{\prime }_{c,d}=0{\prime }_{c,d}\circ 0_{c,c}=0_{c,d}$$0'_{c,d} = 0'_{c,d} \circ 0_{c,c} = 0_{c,d}

for any $c,d$. Thus, the existence of zero morphisms can be regarded as a property of a category, rather than structure on it. (To be more precise, it is an instance of property-like structure, since not every functor between categories with zero morphisms will necessarily preserve the zero morphisms, although an equivalence of categories will.)