Assume $\mathcal{C}$ is a monoidal category, with unit $I$. Given a monoid object $M$, I'd like to talk about modules over $M$, but couldn't find any reference. This might seem quite a stretch, but it is not so bad:

- there are notions of left, right and bimodule that "play well" with residuals
- modules of the same "handedness" make up categories as we would expect
- all objects are modules over $I$, and
- we can still define the tensor product of right $M$-module with a left $M$-module as a coequalizer (as in this nlab page).

Let's assume all such coequalizers exist in $\mathcal{C}$.

Now, fix monoids $A, B, C: \mathcal{C}$, an $A$-$B$-bimodule $X$ and a $B$-$C$-bimodule $Y$: can we endow $X \otimes_{B} Y$ with an $A$-$C$-bimodule structure as we usually do in the case where $\mathcal{C} = \mathbf{Ab}$?

It seems to me that it should be possible, and indeed by the universal property of coequalizers we can show there are morphisms

$$A \otimes (X \otimes_{B} Y) \leftarrow (A \otimes X) \otimes_{B} Y \rightarrow X \otimes_{B} Y$$ $$(X \otimes_{B} Y) \otimes C \leftarrow X \otimes_{B} (Y \otimes C) \to X \otimes_{B} Y$$

Is this the wrong approach, or should I just *assume* the left arrow is an isomorphism (meaning I'll only consider monoidal categories where $\otimes$ behaves this way)? Is there some reference where this all has already been treated?