Let $C$ be a smooth projective curve over a field $k$ and let $D$ be a divisor on $C$. I have seen people considering a zero-cycle on the surface $C \times C$ which they denote by $D \times D$.
If $D=\sum_{P \in S} n_P [P]$, my understanding is that $D \times D$ is the zero-cycle $\sum_{P, Q \in S} n_P n_Q [(P, Q)]$.
1) Is that correct?
2) Is there a way to write this using the projections $\mathrm{pr}_i \colon C \times C$ to $C$, something like $\mathrm{pr}_1^\ast D \cdot \mathrm{pr}_2^\ast D$?