I was puzzled by this question as well a while ago. Here is a suggestion, which worked for me:
The product and the coproduct of the category are usually defined by an universal property.
The adeles $\mathbb{A}$ are inductive limit of all $S$-ad{'e}les $\mathbb{A}(S)$ for a finite set of places, so they universal property is given as: If you are given a map $\phi : \mathbb{A} \rightarrow X$ to some topological ring, that there exists for every large enough set $S$ an unique $\phi_S : \mathbb{A}(S) \rightarrow X$ such that $\phi_S = \phi$ on $\mathbb{A}(S)$.
Remark: For this to work, it is important all but finitely many compact rings are actually open, since you have to choose the characteristic functions at these points to construct $\phi_S$. So I doubt that the restricted product can be defined in the category of locally compact groups/rings/spaces. But on the other hand, the restricted product can be defined for a family of pairs of spaces $(A_i \supset B_i)_i$ with almost all $B_i$ are open in $A_i$.