Algebra of matrix coefficients in the dual of a hopf algebra, confusing verification Let $H$ be a Hopf-algebra, and let $V$ be a finite dimensional $H$-module (a module for the algebra structure of $H$).
For $f \in V^*$ and $v \in V$, we get $c^V_{f,v} \in H^*$ via $c^V_{f,v}(u) = f(u.v)$.
These are apparently called matrix coefficients, a keyword that google doesn't know much about.
The text I am reading claims:
$c^V_{f,v} c^W_{g,w} = c^{V \otimes W}_{f \otimes g, v \otimes w}$
But this is not what I get:
$c^V_{f,v} c^W_{g,w}(u) = f(u.v) g(u.w)$
while
$c^{V \otimes W}_{f \otimes g, v \otimes w}(u) = f \otimes g( \Delta(u) . v \otimes w)$, where $\Delta : H \to H \otimes H$ is the comultiplication, and $f \otimes g (v \otimes w) = f(v) g(w)$. 
It's not necessarily the case that the comultiplication has $\Delta(u) = u \otimes u$, so I don't see how to conclude. Maybe I am misunderstanding something?
I would appreciate it very much if someone familiar with this material could clarify this.
 A: For $x \in H$ with $x = \sum_{i=1}^n x^{(1)}_i \otimes x^{(2)}_i$ we have
\begin{align*}
 c^{V \otimes W}_{f \otimes g, v \otimes w}(x)
&= m_k\left((f \otimes g)(x.(v \otimes w))\right)
 = m_k\left((f \otimes g)\left( 
    \sum_{i=1}^n (x^{(1)}_i.v) \otimes (x_2^{(2)}.w)
   \right)\right) \\
&= \sum_{i=1}^n f(x^{(1)}_i.v) g(x^{(1)}_2.w)
 = \sum_{i=1}^n c^V_{f,v}(x^{(1)}_i) c_{g,w}(x^{(2)}_i)
 = (c^V_{f,v} c^W_{g,w})(x)
\end{align*}
where $m_k \colon k \otimes k \to k$ denotes the multiplication of the ground field.
You have to be careful with the product $c^V_{f,v} c^W_{g,w}$, as this is not taken pointwise: Because $H$ is in particular a coalgebra the dual $H^*$ naturally carries an algebra structure with the multiplication
$$
 H^* \otimes H^*
 \hookrightarrow (H \otimes H)^*
 \xrightarrow{\Delta^*} H^*,
$$
which gives us for all $f,g \in H^*$ and $x \in H$ with $\Delta(x) = \sum_{i=1}^n x^{(1)}_i \otimes x^{(2)}_i$ that
$$
 (fg)(x) = \sum_{i=1}^n f(x^{(1)}_i) g(x^{(2)}_i).
$$
(As already pointed out this product on $H^*$ also coincides with the convolution product of $\mathrm{Hom}(H,k)$ where we take $H$ as a coalgebra and $k$ as an algebra.)
