In fact, an enrichment over pointed sets consists precisely of the choice of a ‘zero’ morphism for each pair of objects, with the property that and for any morphism . Such an enrichment is unique if it exists, for if we are given a different collection of zero morphisms , we must have
for any . 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.)
See at zero object for examples.