Game Theory: T-Fold Repetition Game
You mostly got it, there are some minor mistakes.


You cannot have the moving index $t$ of the Cartesian product in the superscript of said product
$$S_i={\sf X}^{t-1}_{t=1} A_i^{A^t}.$$
At the start there have been no moves, so in round 1 you should map the empty history $H_0 = A^0$ to $A_i$, not $H_1 = A^1$ as you seem to have done.


Considering these I think the strategy set of $i$ is
$$
S_i={\sf X}^{T-1}_{t=0} A_i^{A^t}.
$$