@HarrisonGrodin Levy and Staton POPL paper develops this for pre-multicategories, I guess a slightly simpler development for multicategories would work.
Multicategories have a Yoneda structure (Hermida develops that one) that would give you one class. You can try finding a Yoneda structure for co-multicategories, it might give you the other. But there's something funny happening for those, so there might be more than one yoneda structure, or maybe none at all and you'd need to work with proarrow equipments.
Just an intuition!
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