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We investigate large cardinal axioms beyond the level of ω-huge in context of the universality of the suitable extender models of [Suitable Extender Models I, J. Math. Log.10 (2010) 101–339]. We show that there is an analog of ADℝ at the level of ω-huge, more precisely the construction of the minimum model of ADℝ generalizes to the level of Vλ+1. This allows us to formulate the indicated generalization of ADℝ and then to prove that if the axiom holds in V at a proper class of λ then in every suitable extender model, the axiom holds at a proper class of λ (provided the relevant ω-huge embeddings can be chosen to preserve the suitable extender model).

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