In the proof of 3.2 in "Notes on Condensed Mathematics", Scholze writes that there is a natural map $\alpha$ of topoi from the topoi of sheaves on compact Hausdorff spaces over a fixed S, and the topos of sheaves on open sets in S. What's that map?
After some deliberation, my guess is that the left adjoint is induced by the map that takes an open set in $S$ to the corresponding condensed set (given by taking continuous maps into $U$), in particular is not induced by a morphism of sites.
I think that this extends to a cocontinuous functor is akin to saying that if $U_{i}$ is an open covering of a $U$ (in $S$), then locally (in the finite-surjection topology) any map from a profinite set into $U$ factors through one of the $U_{i}$.
