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Better formatting and linking to the abstract page: Strict quantum 2-groups
I assume that you're asking why he needs the equation
$h_{(1)} \otimes h_{(2)} \triangleright a = h_{(2)} \otimes h_{(1)} \triangleright a$(for $h$ in the ground Hopf algebra $H$ and $a$ in the Hopf $H$-algebra $A$), rather than the entire set-up in that condition; without that set-up (consisting of $H$ and $A$ themselves), you'd have nothing to talk about.
I don't remember enough about Hopf algebras to answer your question either, but I'm trying to understand it. ☺️
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