Kolda et al. and De Lathauwer et al. talk about the mode-$n$ tensor-matrix product where if $X,Y \in \mathbb{R}^{N \times N \times N}$ and $C \in \mathbb{R}^{N \times N}$, we have \begin{equation*} Y = X \times_n C \end{equation*} such that $Y_{(n)} = CX_{(n)}$ where $Y_{(n)}, X_{(n)}$ are mode-$n$ unfolded forms of $X$ and $Y$.
What kind of formalism do I need to be able to refer to a product such that $Y_{(n)} = X_{(n)}C$ i.e. to describe a product in which the matrix is the right hand operand?
Any help will be greatly appreciated, thanks in advance.