In the Berkeley lecture notes about p-adic geometry, during the fourth lecture, the following claim is made:
Let $S$ be a profinite set and call $\hat{S}$ the affinoid space defined by the ring $C=C^{0}(S,\mathbb{Z})$, after giving It the discrete topology.
Then, $\hat{S}$ represents the functor sending an adic space $X$ to $Hom_{Top}(|X|,S)$, where $|X|$ Is the topological space underlying $X$.
I do not understand why this Is true.
I understand that it is sufficient to show this for $X=Spa(A,A^{+})$ an affinoid space. In this case we have that $Hom_{adic}(X,S)=Hom_{ring}(C,A^{+})$. But worringly, I do not know how to conclude or whether this should now be obvious. Furthermore, I do not see how to use that $S$ Is a profinite space.
Any help, hint or reference would be useful. Especially if somebody could point out whether this should actually be trivial.