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Tim basically answered your question, but let me add: The way Lurie (and others) deal with the yoneda embedding is by abusing the Straightening/Unstraightening equivalence. The gist is that the functor $Map(X,-)$ corresponds to the cocartesion fibration $C_{X/} \rightarrow C$, and the bifunctor $Map(-,-)$ corresponds to $Tw(C) \rightarrow C^{op} \times C$, where $Tw(C)$ is the twisted arrow category. All of this is developed in HTT. (As a more digestible resource I can recommend the notes of Fabian Hebestreit)
This is a wonderful answer. I'm torn about accepting it, since there might still be someone else coming up with useful conditions for a site. The one "problem" I have with Leray sheaves is that it doesn't seem so straightforward to show that the categories I have in mind are in fact given as such Leray sheaves.
@MarkusZetto: Yes, I am. I should have mentioned their paper in the question. They show that topoi that are compactly assembled are equivalently categories of Leray sheaves. (The datum necessary for this notion is given by a category with finite limits and a cocontinuous idempotent monad) But since it is a property of a site that its category of sheaves is compactly assembled, not a structure, I believe there should also be more straightforward characterizations.
@IvanDiLiberti: Yes. Unfortunately, that won't cover many important examples. E.g. categories of sheaves on a space are almost never compactly generated (= locally finitely presentable).
I am not quite sure where it comes from. The terminology is used quite heavily in the K-theory community. Names one should mention are probably Efimov, Nikolaus, Clausen and Scholze, among others of course.
For future readers: It was established on discord that there are two working definitions of $\kappa$-sifted: - Either the weak version, as defined by @Z. M above in terms of the diagonal functors being cofinal. - The strong version, as used by Adámek et al. defining $C$ to be $\kappa$-sifted if $C$-indexed colimits in Sets (1-categorically) or Spaces ($\infty$-categorically) commute with $\kappa$-small products. The weak version does not need to imply the strong version - This is where the confusion comes from.
