Let $C_1$ and $C_2$ be smooth, projective curves over a field $K$. Let $S = C_1 \times C_2$. Let $D$ and $D'$ be (reduced) divisors on $S$ which map dominantly to both $C_1$ and $C_2$.
How does one calculate $(D \cdot D')$ from the degrees of $D$ and $D'$ relative to $C_1$ and $C_2$? Does the classical formula for $\mathbf{P}^1 \times \mathbf{P}^1$ remain valid, i.e. if $D$ is a curve of bidegree $(m,n)$ and $D'$ is a curve of bidegree $(m',n')$, then $(D \cdot D') = mn' + nm'$?