On projections of the tails of a power

Let 𝜅 be an inaccessible cardinal, 𝔘 a universal algebra, and ∼ \sim the equivalence relation on U κ \mathfrak{U}^{\kappa} of eventual equality. From mild assumptions on 𝜅, we give general constructions of E ∈ End ( U κ / ∼ ) \mathcal{E}\in\operatorname{End}(\mathfrak{U}^{\kappa}/{\sim}) satisfying E ∘ E = E \mathcal{E}\circ\mathcal{E}=\mathcal{E} which do not descend from Δ ∈ End ⁡ ( U κ ) \Delta\in\operatorname{End}(\mathfrak{U}^{\kappa}) having small strong supports. As an application, there exists an E ∈ End ( Z κ / ∼ ) \mathcal{E}\in\operatorname{End}(\mathbb{Z}^{\kappa}/{\sim}) which does not come from a Δ ∈ End ⁡ ( Z κ ) \Delta\in\operatorname{End}(\mathbb{Z}^{\kappa}) .