if $Y = \mathfrak{a}$ and $X_i = \mathfrak{a}$ for all $i$ then: the set of linear orders on $n$ elements, equivalently the elements of the symmetric group$\Sigma_n$;

if $Y = \mathfrak{n}$ and exactly one of the $X_i = \mathfrak{n}$ then: the set of linear order $\{i_1 \lt \cdots \lt i_n\}$ such that $X_{i_n} = \mathfrak{n}$;

given by labeling the unique object/color of $Assoc$ with $\mathfrak{a}$. For $(A,N) \colon LM^\otimes \to \mathcal{C}^\otimes$ a map to a symmetric monoidal category the composite