On projections of the tails of a power

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