We all know that



> 
> A monoidal category is a bicategory with one object.
> 
> 
> 


How do we fill in the blank in the following sentence?



> 
> A multicategory is a ... with one object.
> 
> 
> 


The answer is fairly clear: it'll be a bit like a bicategory, but instead of being able to compose $1$-cells straightforwardly, all we can do is specify, for any 'composable' sequence
$$
A\_0 \xrightarrow{f\_1} \cdots \xrightarrow{f\_n} A\_n\,,
$$
of $1$-cells and any $1$-cell $g \colon A\_0 \to A\_n$, the set of $2$-cells
$$
f\_1,\dots,f\_n \Rightarrow g\,.
$$
What is the name of such a thing? A natural example, for a given SMCC $\mathcal V$, has $\mathcal V$-enriched categories as the objects, profunctors $\mathcal C^{op}\otimes \mathcal C\to \mathcal V$ as the $1$-cells and, for any collection
$$
\mathcal C\_0,\cdots,\mathcal C\_n
$$
of categories and profunctors $F\_i\colon \mathcal C\_{i-1}^{op}\otimes \mathcal C\_i\to \mathcal V$, $G\colon \mathcal C\_0^{op}\otimes \mathcal C\_n \to \mathcal V$, the set of extranatural transformations
$$
\phi\_{b\_1,\cdots,b\_{n-1}} \colon F\_0(a,b\_1) \otimes \cdots \otimes F\_n(b\_{n-1},c)\Rightarrow G(a,c)
$$
as the $2$-cells $F\_1,\cdots,F\_n \Rightarrow G$.


Of course, if $\mathcal V$ is cocomplete, then this is equivalent to the usual bicategory of $\mathcal V$-enriched profunctors.



A multicategory is a ... with one object?
This has been called a "fc-multicategory" by Tom Leinster, for example [here](https://arxiv.org/abs/math/9903004). 


I think this as also been called a "Hypervirtual double category" [here](https://arxiv.org/pdf/1511.04070.pdf), but I don't remember if this is exactly the same notion or if there are some additional assumption in the second link.


