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We exhibit a simple combinatorial property which, for a given supercompact cardinal $\\kappa$, characterize the projections of all projections of the strongly compact Prikry forcing using $\\kappa$-complete fine measures. Considering level-by-level results, if $\\kappa$ is $2^\\lambda$-strongly compact, we characterize the forcings of size $\\leq\\lambda$ which are projections of that $\\lambda$-strongly compact Prikry forcing. 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