Is $G^{(X, Y)} = (G^X)^Y?$ ($A^B$ just means that $B$ is mapped to $A$)

So I have a set $G={banana,potato}$.To do $G^{(X,Y)}$ ($A^B$ just means that map $(x,y)$ to $G$),I map a Cartesian product $(x,y)$ to every element of $G$,which produce
${((x,y),banana),((x,y),potato)}$.What if I have $(G^X)^Y$,is it correct if I do $x * y$ first ,then map them to $G$,which produce same element$?$ So is $G^{(X,Y)} = (G^X)^Y?$