I'm reading a proof of the following theorem
If $H$ is a Hopf algebra with invertible antipode then Yetter-Drinfeld modules of finite dimension form a rigid category.
In this proof we define $V^*$ in a natural way as $\mathrm{Lin}_k(V,k)$. We define action and coaction on $V^*$ by $\langle h\rhd f,v \rangle:=\langle f,Sh\rhd v\rangle$ and $\langle\delta(f),v\rangle:=S^{-1}\left(v_{(-1)}\right)\langle f,v_{(0)}\rangle$.
There is one step in this proof which I don't understand, namely :
$S^{-1}\left(\left(Sh\right)_{(1)}v_{(-1)}\right)\left\langle f, \left(Sh\right)_{(2)}\rhd v_{(0)}\right\rangle=\\=S^{-1}\left(\left(\left(Sh\right)_{(1)}\rhd v\right)_{(-1)}\left(Sh\right)_{(2)}\right)\left\langle f, \left(\left(Sh\right)_{(1)}\rhd v\right)_{(0)} \right\rangle$
How to obtain this equality ? (Maybe it is obvious, but I don't know how to do it.)