Coset construction is a well-known method to obtain new vertex algebras from known ones. Recently, Davide Gaiotto had started to give new interpretations of coset construction in terms of 4d gauge theories and the geometric Langlands program. In this lecture I first talk about my joint work with Thomas Creutzig and Andrew Linshaw that proves a long-standing conjecture on the coset construction of W-algebras. Then I explain the method used in this work also proves a conjecture of Dennis Gaitsgory that is crucially used in the quantum geometric Langlands program.