TITLE: Category theory with multisets
QUESTION [9 upvotes]: An alternative to the notion of multiset introduced in Section 2.2 of Aluffi Chapter 0 is obtained by considering sets endowed with equivalence relations; equivalent elements are taken to be multiple instances of elements 'of the same kind'. Define a notion of morphism between such enhanced sets, obtaining a category MSet containing (a 'copy' of) Set as a full subcategory. (There may be more than one reasonable way to do this! This is intentionally an open-ended exercise.) Which objects in MSet determine ordinary multisets as defined in Section 2.2 and how? Spell out what a morphism of multisets would be from this point of view. (There are several natural notions of morphisms of multisets. Try to define morphisms in MSet so that the notion you obtain for ordinary multisets captures your intuitive understanding of these objects.)

REPLY [3 votes]: NOTE: I first thought of posting this as a reply to @JohannasHahn's answer, but it got too long.
At the risk of being too pedantic, I would say that our morphisms must be triplets: $(f,\sim_A,\sim_B)$, since technically sets of morphisms should be disjoint [Aluffi, Def. 3.1 on pg.19]. Alternatively we can define a morphism $(A,\sim_A)\rightarrow(B,\sim_B)$ to be a map $f:(B/\sim_B)\rightarrow(A/\sim_A)$.
The last (and vaguest) part of the question remains unanswered:

Which objects in $\mathsf{MSet}$ determine ordinary multisets as defined in §$2.2$ and how? Spell out what a morphism of multisets would be from this point of view. (There are several natural notions of morphisms of multisets. Try to define morphisms in $\mathsf{MSet}$ so that the notion you obtain for ordinary multisets captures your intuitive understanding of these objects.)

The definition for multisets given in §$2.2$ is a map, $m:A\rightarrow\mathbb N^*$, from a set $A$ to the positive integers. This map can be viewed as a multiset, where every element $a\in A$ appears $m(a)$ times.
I have not been able to answer the above in a satisfactory way with either of the two definitions for morphisms in $\mathsf{MSet}$ we have presented here. Any thoughts?