"Boundaries" in Free Simplicial Monoids

I suspect that this has been addressed somewhere already, but I cannot find anything. Let $hat{Delta}$ denote simplicial sets, and $Monhat{Delta}$ simplicial monoids. There is a forgetful functor $(-):Monhat{Delta}tohat{Delta}$ which admits a left adjoint $-^{*}:hat{Delta}to Monhat{Delta}$. My idea is to define a sort of "boundary" operator as a natural endotransformation of $-^{*}$, whose component at each $X:hat{Delta}$ is the function sending lists of n-cells to the concatenation of the corresponding list of lists of faces of each n-cell. It’s easy to see that this a shifted homomorphism of simplicial monoids. I don’t feel safe calling it a boundary since the free monoid discerns between $(x)$ and $(x,x)$ (these are living in some grade of $X^{*}$), and therefor the boundary of closed structures is captured by sequential redundancies rather than simply vanishing algebraically. I want to keep track of this redundancy rather than quotienting mindlessly, since part of my motivation is to recover the homotopical information which is lost when abelianizing into homology. Using noncommutative monoids maintains harmony with the simplex category that locally models simplicial sets, by allowing the ordering of simplicial faces (associated to cohomotopic orientation via the homotopy correspondence, see the Hopf degree theorem) to interact politely with the synthetic algebra of the concatenation of lists of cells. We exfoliate the structure condensed into a simplicial set $X$ by "dissolving" or analysing the ordered (simplicial) stuff in a bath of freely-orderable (listable) bits. The full picture of this will be lucid given the universal property of $Delta_{alpha}$ as the free "monoidally-pointed category" (monoidal category equipped with a monoid in it) on a point, so perhaps I would be better suited working with augmented simplicial sets to answer my question.

My question is whether this list-of-lists-of-faces-of-lists-of-cells operation is documented anywhere, and whether it has any notable properties to justify its study in the context of (co)homotopy?