1. Context
Let $\mathbb k$ be a field. Let $H$ be a $\mathbb{k}$-Hopf algebra. Let $U, V$ be objects in the category $H\text{-}\mathrm{mod}$ of left $H$-modules. (In particular they are $\mathbb{k}$-vector spaces.)
Then $\mathrm{Hom}_{\mathbb{k}}(U, V)$ can be given the structure of a left $H$-module via
$$
(h \mathbin{.} f)(u) := h_{(1)} \mathbin{.} f( S(h_{(2)}) \mathbin{.} u ) \,.
$$
The ground field $\mathbb{k}$ can be endowed with the structure of a left $H$-module via $$ h \mathbin{.} k := \epsilon(h) \cdot k \,. $$
By combining these actions $U^* = \mathrm{Hom}_{\mathbb{k}}(U, \mathbb{k})$ becomes a left $H$-module.
Using counitality of the Hopf algebra $H$ this left $H$-action on $U^*$ simplifies to the following: $$ (h \mathbin{.} f)(u) := f(S(h) \mathbin{.} u) \,. $$
From now on let $U$ and $V$ be finite-dimensional. Then the categorical right dual of $U$ in $H\text{-}\mathrm{mod}$ (denoted $U^\vee$) is precisely $U^*$ with action given by the above action.
We can define the above $H$-action on $U^\vee$ in terms of string diagrams as follows:
2. Question
- If $U$ and $V$ are finite-dimensional: How do you write down the above $H$-module structure on $\mathrm{Hom}_{\mathbb{k}}(U, V)$ in terms of string diagrams?
3. My thougths
In the finite-dimensional case there is a vector space isomorphism $\mathrm{Hom}_{\mathbb{k}}(U, V) \cong U^* \otimes V$.
Hence, my guess would be as follows:
Is my guess correct? I suppose I would have to translate the algebraic definition via the above isomorphism to show that the graphical and the algebraic definition are equivalent?