Let $\Sigma$ be an $n$-dimensional representation of the global Weil group $W_F$ for a number field $F$, and $\Pi$ an automorphic representation of $\operatorname{GL}_n(\mathbb A_F)$. Suppose that at every place $v$ of $F$ where $\Sigma_v$ and $\Pi_v$ are both unramified, we have $\Pi_v \leftrightarrow \Sigma_v$ under the local Langlands correspondence.

In Guy Henniart's proof on the local Langlands correspondence, he claims that $\operatorname{Det} \Sigma$ then corresponds to the central character of $\Pi$ under the isomorphism $W_F^{\operatorname{ab}} \cong \mathbb A_F^{\ast}/F^{\ast}$ of global class field theory.

Why should this be the case? It is true that at almost all places $v$, $\operatorname{Det} \Sigma_v$ identifies with the central character of $\Pi_v$ under the isomorphism $F_v^{\ast} \cong W_{F_v}^{\operatorname{ab}}$. We have for $a = (a_v) \in \mathbb A_F^{\ast}$,

$$\varpi_{\Pi}(a) = \prod\limits_v \varpi_{\Pi_v}(a_v)$$

so it seems like something could go wrong with $\varpi_{\Pi}$ at the ramified places.