In proposition 2.7. of the condensed notes of professors Scholze and Clausen it is said that the category of extremally disconnected sets is a site, but in the definition of a site in the Stacks Project (https://stacks.math.columbia.edu/tag/00VH) it is necessary for a site to have fibre products sometimes (Axiom (3) in the definition) and extremally disconnected sets don't have all fibre products.
The category of extremally disconnected sets is a site using the definition from the Stacks Project?