Is There a Name for a Subcategory Acting Like an Ideal in Category Theory?

[tmatrix]

Consider a category C with objects ob(C) and morphisms hom(C). Suppose there is a subcategory D such that ob(D)=ob(C) but hom(D) is a subset of hom(C), with the property that the product of two morphisms in hom(C), f*g, is an element of hom(D) if either f or g is in hom(D).

This subcategory is basically acting like an "ideal" in algebra, but I'm not sure what this thing is called in the context of categories. I know nothing more about category theory than the ability to phrase the above question.

Does anyone know what to call it?

 

[fresh_42]

You could call it a normal subcategory, but I do not think there is a special name for it. The usual properties for subcategories are "full" or "regular". I wouldn't bet that "normal" isn't occupied either. Have a look:
https://ncatlab.org/nlab/show/full+subcategory