The Catalan simplicial set II

The Catalan simplicial set $\mathbb{C}$ is known to classify skew-monoidal categories in the sense that a map from $\mathbb{C}$ to a suitably defined nerve of $\mathrm{C}\mathrm{a}\mathrm{t}$ is precisely a skew-monoidal category \cite{Catalan1}. We extend this result to the case of skew monoidales internal to any monoidal bicategory $\mathcal{B}$. We then show that monoidal bicategories themselves are classified by maps from $\mathbb{C}$ to a suitably defined nerve of $\mathrm{B}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}$ and extend this result to obtain a definition of skew-monoidal bicategory that aligns with existing theory.