1. Context
Let $H$ be a Hopf algebra over a field $\mathbb k$. Let $(V, p)$ be a finite dimensional (left) $H$-module. We want to endow its dual vector space $V^*$ with the structure of a (left) $H$-module. For that end define the map $$ p' \colon H \xrightarrow{\enspace S \enspace} H \xrightarrow{\enspace p \enspace} \operatorname{End}(V) \xrightarrow{\enspace (-){^*} \enspace} \operatorname{End}(V^*) , $$ where $(-)^* \colon \operatorname{End}(V) \rightarrow \operatorname{End}(V^*)$, $f \mapsto f^*$.
In a strict monoidal category we have a graphical calculus. In the following we consider a strictification of $\mathrm{vect}_{\mathbb k}$. Then one can write down in string diagrams the definition of the above (left) $H$-action on $V^*$ as follows:
2. Questions
- This picture seems to show that the braiding enters in the definition of the induced (left) $H$-module structure on $V^\vee$ from the (left) $H$-module structure on $V$. Correct? How so?
It is correct, why are you surprised by this?
By the way, notice that you don't need finite-dimensionality of $V$. Since $$\mathsf{Hom}(H \otimes V^*,V^*) \cong \mathsf{Hom}(H \otimes V^* \otimes V, \Bbbk)$$ by the hom-tensor adjunction, you may describe the action of $H$ on $V^*$ resorting to string diagrams as the morphism uniquely determined by
(BTW: sorry, my diagrams go from top to bottom)