ct.category theory – the category of multisets and spans

I've been thinking about multisets for a while. These are sets where the elements can be repeated, so that $ S = {a, a, b, c, b } $ it's a multiset on the set $ A = {a, b, c } $.

I've also been looking at morphisms among multisets. Take two multisets $ S_A, S_B $ with underlying sets $ A, B $. I would like to define a map between multisets. $ S_A, S_B $ as a lapse in the underlying sets, so that $ f = A leftarrow C igh B $Y $ f: S_A rightarrow S_B $.

This is how I am defining the morphisms. Take a multiset $ S_A $ and let $ a_i $ be one of the terms, equally for $ S_B $ Y $ b_j $. We have an indexing set. $ C $ and let the $ f, g $ be arms of such a span $ f: C rightarrow A $ Y $ g: C rightarrow B $. Leave $ c_i in C $ and let $ f (c_i) $ be the established element of the term $ a_i $equally for $ g (c_j) $. This makes sense to me in terms of three columns in a database, $ A, B, C $ where $ C $ It's like a primary key.

I'm not sure how to define the composition of span. I'm thinking a lot about David Spivak's work these days, so I'll suggest a definition of his text, see section 2.5.2.3, which I can not reproduce here. He uses fiber products.

Does my definition of objects and morphs define a category?