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?
