Let $ F$ be a local field of characteristic 0. The main theorems in local class field theory can be summarized by the existence of a group $ W_F$ and a map $ $ \phi_F:W_F\to W_F^\mathrm{ab}\simeq F^\times $ $ satisfying certain properties. Let $ \Pi(W_F)$ and $ \Pi(F^\times)$ denote the characters of $ W_F$ and $ F^\times$ . Then $ \phi_F$ induces a bijection between $ \Pi(W_F)$ and $ \Pi(F^\times)$ that preserves the associated $ L$ -functions and $ \varepsilon$ -factors. Call this bijection $ \psi_F$ . This is basically the local Langlands classification for $ \mathrm{GL}(1)$ .

My question is, to what extend is the bijection $ \psi_F$ between $ \Pi(W_F)$ and $ \Pi(F^\times)$ unique? In other words, suppose one has a bijection $ \eta$ from $ \Pi(W_F)$ to $ \Pi(F^\times)$ that preserves $ L$ -functions and $ \varepsilon$ -factors. Is $ \eta$ equal to $ \psi_F$ ? Are there any other properties in addition to preserving $ L$ and $ \varepsilon$ that characterize $ \psi_F$ uniquely?

I’m also interested in global solutions. If $ F$ is now a number field, is the family of bijections $ $ \psi_F:\Pi(W_F)\simeq \Pi(F^\times\backslash \mathbb{A}_F^\times) $ $ and $ $ \psi_{F_v}:\Pi(W_{F_v})\simeq\Pi(F_v^\times) $ $ induced by $ \phi_F$ and $ \phi_{F_v}$ somehow unique?