For that i defined the map $\mathcal{O}_{X\times X}\to \mathcal{O}_X\times \mathcal{O}_X$, $[(x,y)] \mapsto ([x],[y])$

where $O_Z$ denotes the set of orbits of $Z$ and $[z]$ denotes the orbit $\{g\cdot z : g\in G\}$

I want to check that this map is well-defined and surjective.

Please hepl me