Let $Gr$ be the affine Grassmannian of $G=G((t))/G[[t]]$, and let $Perv(Gr)$ be the category of perverse sheaves on $Gr$. We have action of $G((t))$ on the left-hand side of $Perv(Gr)$, also we have action of the tensor category $Rep(G^\vee)$ on the right-hand side, through geometric Satake correspondence. 
It is clear that we have the action of $G((t))$ on $Perv(Gr)$ as functors.


Follow Gaitsgory's paper "the notion of category over stack", he claimed that an action of algebraic group scheme $H$ on a category $\mathcal{C}$ is actually equivalent to a category $\mathcal{C}$ with the action of the tensor category "Rep(H)".


Go back to the original set-up, that means we also have an action of $G^\vee$ on $Perv(Gr)$.
I think, on this $Perv(Gr)$, the actions of $G((t))$ and $G^\vee$ should have different meaning, but why they gave them the same name?


Am I confused?



A question on group action on categories 
Yes, you are confused. What is claimed by Gaitsgory is that datum of category with
action of $H$ is equivalent to datum of of another category with action of $Rep(H)$.
You go back and forth between these two categories using constructions of "equivariantization"
and "de-equivariantization".


