Recall that a monoidal category $\newcommand{\C}{\mathcal{C}} (\C,\otimes,I)$ comes equipped with an associator $\alpha_{XYZ} \colon X \otimes (Y \otimes Z) \xrightarrow{\sim} (X \otimes Y) \otimes Z$ as well as left and right unitors $\lambda_X \colon I \otimes X \xrightarrow{\sim} X$ and $\rho_X \colon X \otimes I \xrightarrow{\sim} X$ (where $X$, $Y$, and $Z$ are objects of $\C$). Then $\C$ is said to be strict if the associator and left and right unitors are all identity morphisms.
Question: Is there a term for the situation where the associator is the identity, but where the unitors need not be identities?
The reason I ask is because I typically treat tensor products of $R$-modules (where $R$ is a commutative ring) in this way. I think of there being a uniquely determined triple tensor product $L \otimes_R M \otimes_R N$ so that there is essentially no difference between the two different ways to associate the triple tensor. However, I tend to think of $R \otimes_R M$ as being only naturally isomorphic to $M$, not equal to it.
