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In this work we present a phase-space analysis of the cosmological relaxation of generalized gravity theories where the gravitational constant G varies towards general relativity. We assess the interplay of the two main mechanisms that yield general relativity as the cosmological attractor: (i) the vanishing of the coupling function α(φ), and (ii) the existence of a minimum of the scalar field potential. The latter mechanism is shown to supersede the first. We classify the fixed points associated to different types of potentials and discuss the late time self-similar solutions that arise when the scalar field potential exhibits an asymptotic exponential behaviour from the viewpoint of the relaxation mechanism.