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The PSI Positron Production (P^{3} or P-cubed) experiment is a e^{+} source and capture system with the potential to increase by an order of magnitude the state-of-the-art e^{+} yield normalized to the drive linac energy, a highly desirable goal for future colliders. The experiment is framed in the FCC-ee injector study and will be hosted at SwissFEL, located at the Paul Scherrer Institute in Switzerland. This paper presents the P^{3} project at an advanced stage, with an emphasis on a capture system featuring a novel e^{+} matching device based on high-temperature superconducting solenoids, followed by two large aperture rf cavities surrounded by normal-conducting solenoids. The diagnostics design is also introduced, including monitors of charge, energy spectrum, and bunch-by-bunch longitudinal profile simultaneously for secondary e^{+} and e^{-}. The last chapter of the text overviews the currently ongoing installation at SwissFEL, including the beam transfer line, rf network, radiation protection, and other relevant activities toward the operation with e^{+} in the coming years.