Let $\{E_{ij}:(i,j)\in I\times J\}$ be a family of locally convex Hausdorff spaces. We study the relation between the spaces $P_{(i,j)\in I\times J}(E_{ij})$ and $P_{j\in J}(P_{i\in I}(E_{ij}))$. In particular, we prove that the ultraproduct of an ultraproduct is again an ultraproduct.