Free monodromic Hecke categories and their categorical traces

The goal of this paper is to give a new construction of the free monodromic categories defined by Yun. We then use this formalism to give simpler constructions of the free monodromic Hecke categories and then compute the trace of Frobenius and of the identity on them. As a first application of the formalism, we produce new proofs of key theorems in Deligne--Lusztig theory.