If K is a field, v a discrete valuation, and k the residue field, there is a residue map $\partial: K^M_n(K) \to K^M_{n - 1}(k)$. All the definitions I have seen for this map involve two pages of combinatorics of different cases (e.g. the book of Fesenko Vostokov).

Does anyone know of a clean, concise, compact definition of this map, for example via some universal property that it might satisfy?