For $G$ a group with action on two objects$X$ and $Y$, there is a canonically induced action on the product$X \times Y$. This is the diagonal action.

Definition

For every $g\in G$, $g:X\rightarrow X$, $x\mapsto gx$, and $g:Y\rightarrow Y$, $y\mapsto gy$ are both automorphisms, and the maps $g:X\times Y$, $g(x,y)=(gx, gy)$ define the diagonal action of $G$ on $X\times Y$.