I have a naive question about the difference of Artin reciprocity in the number field versus function field case: In the number field case, the double quotient we look at is $$K^\times \backslash \mathbb{I}_K / \mathcal{O}$$ ($\mathcal{O}$ denoting the connected component) where as in the function field case we consider $$K^\times \backslash \mathbb{I}_K / \prod_v\mathcal{O}_v^\times.$$ My question is to the relation between the two double quotients, or equivalently to the relation between $\mathcal{O}$ and $\prod_v\mathcal{O}_v^\times$.
Is there some natural (dense) containment? Do these two double quotients "talk" to each other and does this explain why Artin reciprocity has a different form for number fields vs function fields?
