The module category of a bialgebra and its tensor action

Let $k$ be a field and let $H$ be a $k$-vector space with associative unital multiplication $m$, unit $1_H$, coassociative comultiplication $\mathrm{\Delta }:H\to H{\otimes }_{k}H$, and counit $\epsilon :H\to k$. Assume $\mathrm{\Delta }$ and $\epsilon$ are unital algebra maps; this is the bialgebra interface used on this page. A left $H$-module is a unital left module in the sense of Unital left and right modules over a ring; unqualified module means left module: its action satisfies $1_{H}v=v$ and $(h{h}^{\mathrm{\prime }})v=h(h^{\mathrm{\prime }}v)$. Let $H\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$ be the category whose objects are all left $H$-modules and whose morphisms $f:V\to W$ are $k$-linear maps satisfying $f(hv)=hf(v)$ for every $h\in H$ and $v\in V$.

For left $H$-modules $V,W$, give the algebraic tensor product $V{\otimes }_{k}W$ the candidate action

$h\cdot (v\otimes w)=\sum h_{(1)}v\otimes h_{(2)}w,\mathrm{\Delta }(h)=\sum h_{(1)}\otimes h_{(2)}\mathrm{.}$

The sums are finite algebraic tensor sums; the tensor product and scalar conventions are those of Scalars, tensor powers, the empty tensor, opposite algebras and finite sums. The candidate unit object is $k_{\epsilon }=k$ with $h\cdot c=\epsilon (h)c$. For morphisms $f:V\to V^{\mathrm{\prime }}$ and $g:W\to W^{\mathrm{\prime }}$, the candidate tensor morphism is the usual linear map $f\otimes g$ determined by $(f\otimes g)(v\otimes w)=f(v)\otimes g(w)$ through Universal property of the tensor product for balanced maps into abelian groups.

These formulas specify the proposed tensor operations; Bialgebra module tensor actions and their coherence ↗ proves that they are module actions and that the canonical associator and unit maps are morphisms in $H\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$.