I was puzzled by this question as well a while ago. Here is a suggestion, which worked for me:
The product and the co-product of categories are best defined by an universal mapping property.
The adeles $\mathbb{A}$ are the inductive limit of all $S$-adeles $$\mathbb{A}(S) = \mathbb{R} \times \prod\limits_{p \in S} \mathbb{Q}_p \times \prod\limits_{p \notin S} \mathbb{Z}_p$$ for a finite set of places. The universal property is described as follows: If you are given a map $$\phi : \mathbb{A} \rightarrow X$$ to some topological space $X$, then there exists for every large enough set $S$ a 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 that all but finitely many compact subrings ($\mathbb{Z}_p$ in the above context) are actually open subrings, since you have to choose the characteristic functions of this set at the places $p \notin S$ to construct $\phi_S$. So I doubt that the restricted product can be defined in the category of locally compact groups/rings/spaces in such a fashion. On the other hand, the restricted product can be defined for a family of pairs of topological spaces $(A_i \supset B_i)_i$ with almost all $B_i$ are open in $A_i$.
