Causal diamonds are object that arise naturally in Mathematical Relativity. However, the structure of the set of all causal diamonds of a given spacetime remains unexplored for the most part. In light of the rise of the use in Relativity of synthetic methods inspired on metric geometry, being able to describe the geometry and geodesic structure of such a hyperspace -when endowed with its associated Hausdorff distance- is a natural first step towards the study of stability results associated to metric convergence. In this work we furnish the hyperspace of causal diamonds of a Lorentzian product -RXS with a Lorentzian pre-length space structure. We further show that for complete S such space is globally hyperbolic and provide a explicit description of a family of its maximal timelike curves. Finally, we establish a distance and 1-homothety map between this hyperspace and a Lorentzian uniform product. This is joint work with W. Barrera and L. Montes.