On projections of the tails of a power

Let $κ$ be an inaccessible cardinal, $\mathfrak{U}$ be a universal algebra, and $\sim$ be the equivalence relation on $\mathfrak{U}^κ$ of eventual equality. From mild assumptions on $κ$ we give general constructions of $\mathcal{E} \in End(\mathfrak{U}^κ/\sim)$ satisfying $\mathcal{E} \circ \mathcal