The Satake isomorphism plays a basic role in the Langlands program. Gaitsgory and Lurie have formulated a conjectural analog for quantum Langlands, known as the Fundamental Local Equivalence. After reviewing in detail some background and motivation, we describe aspects of the proof of a tamely ramified variant, also conjectured by Gaitsgory. As a corollary, we obtain the Fundamental Local Equivalence over a point. This is a report on joint work with Justin Campbell and joint work in progress with Sam Raskin.