TITLE: A multicategory is a ... with one object?
QUESTION [18 upvotes]: 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.

REPLY [5 votes]: I think it worth mentioning that precisely the notion described in the question is given in Cockett–Koslowski–Seely's Morphisms and modules for poly-bicategories, where it is called a multi-bicategory. A one-object multi-bicategory is a multicategory. As noted in the comments for Simon Henry's answer, Leinster's fc-multicategories are strictly more general: they are to multi-bicategories what pseudo-double categories are to bicategories. A multi-bicategory is precisely an fc-multicategory whose vertical cells are all identities.