Find conjugacy classes of $S_3 \times S_3$
So I have that $S_3$ has conjugacy classes $\{e\},\{(12),(23),(13)\},\{(123),(132)\}$
And $(\pi_1,\pi_2)$ is conjugate to $(\sigma_1,\sigma_2)$ iff there exists a $(\phi_1,\phi_2)$ such that $(\pi_1,\pi_2)=(\phi_1,\phi_2)(\sigma_1,\sigma_2)(\phi_1^{-1},\phi_2^{-1})$
Which I believe tells me that I should have $9$ total conjugacy classes, since this gives me $(\pi_1,\pi_2)$ is conjugate to $(\sigma_1,\sigma_2)$, iff $\phi_1$ is conjugate to $\sigma_1$ and $\phi_2$ is conjugate to $\sigma_2$. So I have $3$ choices of conjugacy class in the first argument and $3$ in the 2nd, so I get $3^2=9$ total possible classes?