Let $B = (B, \nabla, \eta, \Delta, \epsilon )$ be a bialgebra over a commutative ring $k$. Let $M$ and $N$ be two left $B$-modules. Then the tensor product $M \otimes_k N$ becomes a left $B$-module with multiplication rule given by
$$ B \otimes M \otimes N \xrightarrow{\Delta \otimes 1} B \otimes B \otimes M \otimes N \xrightarrow{1 \otimes \tau \otimes 1} B \otimes M \otimes B \otimes N \xrightarrow{\mu \otimes \mu} M \otimes N$$
where the $\mu$'s are the $B$-module multipication on $M$ and $N$. I am trying to show that $H$-$\mathsf{Mod}$ forms a monoidal category with tensor product $- \otimes_k -$. We have a $k$-linear associator $$ \alpha_{MNP} \colon (M \otimes_k N) \otimes_k P \xrightarrow{\sim} M \otimes_k (N \otimes_k P)$$ and I think this is supposed to be the associator in $H$-$\mathsf{Mod}$, so I need to show it is $B$-linear, i.e. that $$b [m \otimes (n \otimes p)] = \alpha_{MNP} (b[(m \otimes n) \otimes p])$$ but I am not really getting anywhere with it. Does anyone have any advice or at least be able to point me to a souce where this is covered? I am currently using Pareigis's Advanced Algebra, but proving monoidal structure is left as an exercise.
