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Composite Higgs models feature new strong dynamics leading to the description of the Higgs boson as a bound state arising from the breaking of a global (flavour) symmetry. These models generally include light states generated by the same dynamics, the detection of which may present the first observable signs of compositeness. One such state is a pseudo-scalar boson resulting from the breaking of a $U(1)$ symmetry common to most composite setups, and whose hints are expected to be visible through low-mass resonance searches at present and future hadron and lepton colliders. In this work we study the phenomenology of this pseudo-scalar field. We show that, for a light state, bottom quark loop effects dominantly impact the production cross section and considerably modify the decay pattern. Moreover, we make a case for targeted low-mass analyses at future lepton colliders, with an emphasis on high-luminosity machines aiming to operate at low centre-of-mass energies. We present a simplified outline of a search for this light pseudo-scalar at one such machine, considering electron-positron collisions at the $Z$-pole. We focus on a signature arising from the pseudo-scalar decay into a pair of hadronic taus and a production mode association with a pair of leptons of opposite electric charges, and compare cut and count methods with machine learning methods.