Let $C$ be a category, $G$ be a finite group. For each $g \in G$, suppose we have a functor $[g]: C \to C$, such that there are natural equivalences $[g_1] [g_2] \xto{\cong} [g_1 g_2]$, satisfying associativity condition. Then, we can call this a group acting on a category.

Is this picture enough?

Suppose $C = Fuk( [(\C^*)^2]^3)$ and $S_3$ permute the three factors of $(\C^*)^2$. I somehow want the action to preserve the holomorphic symplectic structure, hence the holomorphic volume. Now, this category is very simple, it is just $Loc(T^6)$, and is mirror to $Coh( [(\C^*)^2]^3) $, so we can apply the same construction, we then get $$ Coh( [(\C^*)^2]^n )^{S_n} \cong Coh (Hilb^n( (\C^*)^2)) $$ Now, that's funny, because there are something really non-trivial about the last equivalences. What's the mirror of that statment? $$ Fuk( [(\C^*)^2]^n )^{S_n} \cong ?? $$

I don't know what kind of answer to expect.