To a p-adic formal scheme, one can associate its prismatic or syntomic cohomology. These are cohomology theories that refine the crystalline cohomology of the special fiber and the p-adic etale cohomology of the generic fiber. After these cohomology theories were found by Bhatt-Morrow-Scholze and Bhatt-Scholze, Bhatt-Lurie and Drinfeld gave an interpretation in terms of quasi-coherent cohomology on certain stacks, which, in particular, equips those theories with coefficient systems. In this talk, I will explain how to extend these coefficient systems to well-behaved six-functor formalisms by giving possible interpretations of the stacks of Bhatt-Lurie and Drinfeld as analytic stacks (in the sense of Clausen-Scholze). We will discuss advantages and disadvantages of these different incarnations and why this additional step to analytic stacks is necessary.