Given a scheme $S$ and reasonable topology, such as Zariski, étale, or fppf, is the category of sheaves ${Sh}(({Sch}/S)_{top})$ locally small? Or, if you like to assume that a category must be locally small, are usual sheaf "categories" actually categories at all?
The category $({Sch}/S)$ is not small, so the category of presheaves ${Fun}(({Sch}/S)^{opp},({Sets}))$ is not locally small. In fact this is a sheaf category, using the indiscrete topology (only $\mathrm{id}_X$ is a cover). As pointed out in the comments, the category of sheaves in the the discrete topology (everything, including $\varnothing \to X$, is a cover) is not only locally small, but small. So the answer certainly depends on topology, but it feels like it should be the same between Zariski, étale, and fppf topologies.
At this point I expect a negative answer, but I am still not sure exactly why. Is the Freyd–Street construction of a functor $T$, with endomorphisms indexed by equivalence classes of objects in $(Sch/S)$, an fppf (étale, Zariski) sheaf? $T$ takes an object $X$ to the set of split quotient objects $\pi: X \to Q$, plus a distinct element called $0$: $$TX = \{\text{split quotient objects } \pi:X \to Q\} \amalg \{0\}$$ For a morphism $f: X \to Y$, the function $Tf: TY \to TX$ assigns a quotient object $\pi: Y \to Q$ to the morphism $\pi f: X \to Q$ if $\pi f$ is a split quotient object of $X$, and $0$ if not.
This process of adding a single element to some naturally defined set seems somehow like it should break sheafiness for topologies finer than Zariski. Is that the case? If so, is there a simple counterexample to local smallness of $Sh((Sch/S)_{Zar})$?
This question is relevant: Is the étale site a small category? However, no one there addresses my question directly, and for my purposes I am not interested in small sites.