Consider a weak $n$-category $\mathcal{C}$ for $n \geq 3$. We define a coherence structure on $\mathcal{C}$ to be a collection of higher cells that witness the coherence conditions between composites of $k$-morphisms for $1 \leq k < n$.
Let $\text{Coh}(\mathcal{C})$ denote the poset of coherence structures on $\mathcal{C}$, ordered by inclusion. We say that a coherence structure $S \in \text{Coh}(\mathcal{C})$ is universal if for any other coherence structure $T \in \text{Coh}(\mathcal{C})$, there exists a unique morphism of coherence structures $f: S \to T$ in $\text{Coh}(\mathcal{C})$.
Now, consider the 2-category $\mathbf{nCat}$ of (small) weak $n$-categories, weak $n$-functors, and weak natural $n$-transformations.
Does there exist a 2-functor $U: \mathbf{nCat} \to \mathbf{Cat}$ such that for any weak $n$-category $\mathcal{C}$, $U(\mathcal{C})$ is equivalent to $\text{Coh}(\mathcal{C})$, and moreover, $U(\mathcal{C})$ has an initial object corresponding to a universal coherence structure on $\mathcal{C}$?
If such a 2-functor exists, can we construct it explicitly? If it doesn't exist in general, under what additional conditions on $\mathcal{C}$ or $n$ does it exist?
