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A $T$-multicategory in the sense of Crutwell-Shulman, where $T$ is a monad on a virtual double category $C$ is a monoid in the "horizontal kleisli category", i.e., an object of objects $O$, a horizontal arrow of arrows $A : O \to T O$ with composition and identity cells. Is there a good way to define a co-presheaf on such a generalized multicategory?

This seems to give a natural notion of $T$-presheaf as just a bimodule of the monad with a terminal object, i.e. a horizontal arrow $P : 1 \to T O$ with a composite $P;A \Rightarrow P$ that is compatible with composition and identity. If you look at what this means for specific cases like $T = $ free symmetric monoidal category monad on the virtual equipment of cats, functors and profunctors this looks like the definition of presheaf you would come up with, with $P : 1 \to T O$ giving an abstract notion of map from a list of objects of $O$ to $P$ and so can be used to define universal properties like the product, and it looks like you can use the language of cartesian cells to define the right notion of representability.

On the other had, just taking the dual doesn't look like the right thing. What I'm hoping would happen is that if I try to define the universal property of a coproduct using such a copresheaf I would be "forced" to make it a distributive coproduct. However, if we say a copresheaf is a bimodule $Q : O \to T 1$, this doesn't look right because for our example $T$, this would give us an abstract notion of maps $Q^n \to A$ for each $A \in O$, but it seems to me that the right notion of copresheaf (based on the type theory) would be to give an abstract notion of $A_1,...,Q,...A_n \to B$.

In particular for $A,B \in O$ if I try to define a copresheaf $A+B$ by

$$(A+B)^n \to C = \Pi_{m+l=n} A^m,B^l \to C$$

this only gives me a distributive coproduct-like behavior when $A+B$ repeated is the only thing in the domain, and it looks like for a representing object

$$(A+B),C \to D = (A,C \to D)\times (B,C \to D)$$

will not in general be true.

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