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!
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.