On monoidal Koszul duality for the Hecke category

We attempt to give a gentle (though ahistorical) introduction to Koszul duality phenomena for the Hecke category, focusing on the form of this duality studied in joint work of Achar, Riche, Williamson, and the author. We illustrate some key phenomena and constructions for the simplest nontrivial case of (finite) ${\mathrm{S}\mathrm{L}}_2$ using Soergel bimodules, a concrete algebraic model of the Hecke category.