This question has not yet received a satisfactory answer in my opinion, probably because none was available. The non-symmetric case is essentially trivial, and although Leinster's article is excellent, it has created the misconception that multicategories are somehow part of higher category theory, which they are not.

A multicategory begins with the fundamental premise that:

- When composing a multiarrow with other multiarrows,
- we should be able to do so,
- for each way to choose,
- for each *entry* of the codomain,
- a multiarrow with matching domain.

So the difficulty of composing with a multiset of multiarrows is that (unless the multiset of domains is actually a set), there will in general be more than one way to do that composition. It is hard to deal with this problem without making a mess, and it may even be tempting to appeal to an arbitrary total-ordering of the object set.

I am writing an article about a (new?) definition of multicategory, with many applications and examples in enumerative combinatorics. Comcategories and Multiorders

If people don't have enough karma to reply here, feel free to comment directly on my preprint (use google docs "suggesting" mode).

The difficulty of composing with a multiset of multiarrows is that (unless the multiset of domains is actually a set), there will in general be more than one way to do that composition.