There are two approaches to class field theory that I was taught. The first, is the theory of $L$-functions, Dirichlet characters and so forth (which I described succintly in the question What are the pillars of Langlands? that I asked a long time ago), and the other is through Brauer groups.

To be precise, I was taught that the following sketch is class field theory:

Let $K$ be number field. Let $v$ be any non-archimedean place. Then there is some explicit isomorphism $inv_v:Br(K_v)\rightarrow \mathbb{Q}/\mathbb{Z}$. If $v$ is an archimedean place, then there is an explicit isomorphism $inv_v:Br(K_v)\rightarrow \mathbb{Z}/2\mathbb{Z}$ if $K_v$ is $\mathbb{R}$, and $0$ if $K_v$ is $\mathbb{C}$.

Furthermore, the following sequence is exact:

$1\rightarrow Br(K)\rightarrow \bigoplus_v Br(K_v)\rightarrow \mathbb{Q}/\mathbb{Z}\rightarrow 1$

where the first morphism is obvious, and the last morphism is $\sum_v inv_v$.

I take both of these approaches have a lot of content, but it is not clear how people think of them as equivalent. How does one build a dictionary between the one approach and the other?