In [the paper](http://people.maths.ox.ac.uk/~laugwitz/2010-08%20-%20Yetter-Drinfeld%20and%20Hopf%20Modules%20-%20Bachelorarbeit.pdf), page 28, Definition 4.2.1, the compatibility condition for a Yetter-Drinfeld module over $H$ is 
$$
h\_{(1)} v\_{(-1)} \otimes h\_{(2)}.v\_{(0)} = (h\_{(1)}.v)\_{(-1)}h\_{(2)} \otimes (h\_{(1)}.v)\_{(0)}, v \in V, h \in H.
$$ 
On the other hand, in [the article](https://en.wikipedia.org/wiki/Yetter%E2%80%93Drinfeld_category#Definition), the compatibility condition for a Yetter-Drinfeld module over $H$ is 
$$
\delta(h.v) = h\_{(1)} v\_{(-1)} S(h\_{(3)}) \otimes h\_{(2)}.v\_{(0)}, v \in V, h \in H.
$$ 
Are the two conditions equivalent? Thank you very much.



Compatibility conditions for Yetter-Drinfeld modules
Assume that the second equation holds. Then 
\begin{align\*}
\delta(h\_{1}.v)=(h\_{1}. v)\_{-1}\otimes(h\_{1}.v)\_{0}=h\_{1,1}v\_{-1}Sh\_{1,3}\otimes h\_{1,2}. v\_{0}
\end{align\*}
and therefore
$$
(h\_{1}.v)\_{-1}h\_{2}\otimes(h\_{1}.v)\_{0}=h\_{1,1}v\_{-1}Sh\_{1,3}h\_{2}\otimes h\_{1,2}.v\_{0}=h\_{1}v\_{-1}\otimes h\_{2}.v\_{0}.
$$
Conversely, assume that the first equation holds. Then
\begin{align\*}
 (m\otimes &id)(h\_{11}v\_{-1}\otimes Sh\_2\otimes(h\_{12}.v\_0) )\\ 
 &=(m\otimes id)\left( (h\_{11}.v)\_{-1}h\_{12}\otimes Sh\_2\otimes (h\_{11}. v)\_0 \right)\\
 &=(h\_1.v)\_{-1}h\_2Sh\_3\otimes (h\_1.v)\_0\\
 &=(h.v)\_{-1}\otimes (h.v)\_0.
\end{align\*}


