In the definition of an adic space, usually there is a presheaf defined by first saying what it is on a particular basis of the topology of the underlying space, the so called rational subsets. One then extends this to arbitrary opens by taking the limit over all the rational subsets inside the given open. However, in all the references I looked into so far, it is then quickly pointed out that this is in general not a sheaf, followed by a list of particular cases in which it is one.

In Algebraic Geometry, there are many occasions where some construction does not give a sheaf and one simply forces it to be one by saying "Sheafify!" So the Question is:

-Why doesn't one sheafify the structure presheaf of an adic space? Is there no sheafification functor in this case? If not, what goes wrong?