I am trying to define a category $MSet$ of multisets as sets equipped with an equivalence relation. I will call such objects multisets. The other notion of multisets as pairs $(A,m_A\colon A\to\mathbb{N})$ I will call standard multisets, these are actually multisets in which every equivalence class is finite. I also want the category $Set$ to be a full subcategory of $MSet$.
The obvious way to define a morphism $(A,\sim_A)\to(B,\sim_B)$ is as a map $f\colon A\to B$ such that $\forall a,a'\in A\colon a\sim a'\implies f(a)\sim f(a')$. One thing that bothers me with this definition is that if we consider equivalent elements as indistinguishable from the point of view of the multiset, then in general many set maps would induce "the same" morphism.
Next try is to simply take functions $A/\sim_A \to B/\sim_B$. This does not suffer from the issue above, but I am not satisfied with this either. If we consider the standard multisets then our morphisms are just functions of the underlying sets. What bothers me is that any information on the multiplicity is lost. I'd like for example to be able to define the notion of submultiset, that would come with a natural "inclusion map" that will take multiplicity into account.
The best I could come up with is this:
a morphism $(A,\sim_A)\to(B,\sim_B)$ is a set map $A/\sim_A \to B/\sim_B$ together with a family of inclusions $[a]_{\sim A}\hookrightarrow f([a]_{\sim_A})$.
Is this a good notion of morphism? Is it not too restrictive? I think we could instead ask for bijections $[a]_{\sim A}\to f([a]_{\sim_A})$, but that seems even more restrictive.
This is a problem from Aluffi's Algebra Chapter 0, where the author mentions that this is an "open-ended" exercise. So, I guess there is no single "right" definition of $MSet$.
I did not find a good reference on this topic. I would also like to know what are the applications of the theory of such multisets (I know that standard multisets are abound in combinatorics).