Is there a simple term or standard definition for a category in which each hom set has a commutative monoid structure?

Question

Is there a simple term or standard definition for a category in which each hom set has a commutative monoid structure?

I was working with the category of matrices. But I only needed to use that each hom set has a zero morphism, and that we can add morphisms. So I don't need that every morphism has an addititve inverse.

I found stuff on enriched categories, but it seemed to be overly complicated for the purpose that I was using it for.

Or can I simply define an 'commutative monoid catgory' as a category with zero morphisms, and for each hom set a '+' operation that is associative and commutative, and it has the property that $a + 0 = 0 = 0 + a$?

Are there any foundational issues with such a definition?