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.