Monochromatic homotopy theory is asymptotically algebraic

In previous work, we used an ∞-categorical version of ultraproducts to show that, for a fixed height n, the symmetric monoidal ∞-categories of E_n,p-local spectra are asymptotically algebraic in the prime p. In this paper, we prove the analogous result for the symmetric monoidal ∞-categories of K_p(n)-local spectra, where K_p(n) is Morava K-theory at height n and the prime p. This requires ∞-categorical tools suitable for working with compactly generated symmetric monoidal ∞-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the E_n,p-local ∞-categories.