Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 123 / 4 / 2 / b / Solution

For each place $v$ of $K$ and each $w\mid v$ of $L$, the inclusion $K_v\hookrightarrow L_w$ definesAt all but finitely many finite $v$, the component $x_v$ is a unit, and its image is a unit at every $w\mid v$, so the image is an idele. The map is a homomorphism and is injective because every local inclusion is injective.