Hopf algebra elements whose product with another element is proportional to the counit of that element Let $H$ be a Hopf algebra. $\epsilon$ is the counit. $S$ is the antipode. Let $a' \in H$ be such that
$$
a a' = \epsilon(a)a' \quad\forall a\in H.
$$
Prove that
$$
a'_{(1)} \otimes a a'_{(2)} = S(a) a'_{(1)} \otimes a'_{(2)}.
$$
I have no idea how to prove this, it feels like it should be easy. 
At first, the top condition seemed very close to the condition for H to be a H-module algebra over itself (except with $a'$ replaced with the identity in $H$), but research in Kassler and otherwise led me nowhere. The first line of the left side of the proof also made me think that $a$ is an element of the coinvariants of H, if H again had a module structure over itself (as then its coproduct would be the tensor product but with $a$ on the right slot), and it would be proportional to the counit of $a$ because of this. See example 4.4.5 of the Dascalescu book. Again, couldn't bring the antipode into this despite numerous algebraic tricks using its properties.
Any help would be appreciated!
 A: Such an element in the algebra of functions on a finite quantum group is called the Haar element. Finite quantum groups have multi matrix algebras of functions and as the counit is a character there must be a one dimensional matrix factor. The Haar element is a suitably normalised basis vector for this 1-D subspace, and is $\delta_e$ in the commutative case.
Leaning heavily on Timmermann (Examples 1.3.4) and Van Daele (proof of Lemma 1.2), and denoting $a'=:e_1$:
Lemma $$1_H\otimes a=\sum S(a_{(1)})a_{(2)}\otimes a_{(3)}.$$
Proof: Taken straight from Timmerman,
$$
\begin{aligned}
1_H\otimes a&=\sum 1_H\otimes \varepsilon(a_{(1)})a_{(2)}
\\&= \sum\eta(\varepsilon(a_{(1)}))\otimes a_{(2)}
\\&=\sum S({a_{(1)}}_{(1)}){a_{(1)}}_{(2)}\otimes a_{(2)}
\\&:=\sum S(a_{(1)})a_{(2)}\otimes a_{(3)} \quad \bullet
\end{aligned}$$
Now following Van Daele:
$$\begin{aligned}
(1_H\otimes a)\Delta(e_1)&=\sum \left(S(a_{(1)})a_{(2)}\otimes a_{(3)}\right)\Delta(e_1)
\\ &=\sum (S(a_{(1)})\otimes 1_H)\Delta(a_{(2)})\Delta(e_1)
\\ &=\sum (S(a_{(1)})\otimes 1_H)\Delta(a_{(2)}e_1)
\\&=\sum (S(a_{(1)})\otimes 1_H)\varepsilon(a_{(2)})\Delta(e_1)
\\&=\sum (S(a_{(1)}\varepsilon(a_{(2)}))\otimes 1_H)\Delta(e_1)
\\&=\left(S\left(\sum a_{(1)}\varepsilon(a_{(2)})\right)\otimes 1_H\right)\Delta(e_1)
\\&=(S(a)\otimes 1_H)\Delta(e_1),
\end{aligned}$$
which is what you were looking for.
