Sifting composite from elementary models at FCCee and CePC

New Physics models with either an elementary or composite origin are often associated with a similar imprint in a direct search at colliders, case in point being the production of a light pseudoscalar in association with a monochromatic photon from the decay of a Z boson at future $e+e-$ colliders. We exploit the correlation between the discovery of a signal in the Z decays and electroweak precision measurements as a tool to distinguish a composite model from an elementary scalar one. Our results offer an appealing and rich physics case for future colliders and demonstrate how a lepton collider at the Z mass can be a discovery machine for new physics in the Higgs sector.