Prismatic F-crystals and Wach modules

For an unramified extension $K/\mathbb{Q}_p$ with perfect residue field, by works of Fontaine, Colmez, Wach and Berger, it is well-known that the category of Wach modules over a certain integral period ring $\mathbf{A}_K^+$ is equivalent to the category of lattices inside crystalline representations of $G_K$, i.e. the absolute Galois group of $K$. Moreover, by recent work of Bhatt and Scholze, we also know that lattices inside crystalline representations of $G_K$ are equivalent to the category of prismatic $F$-crystals over $O_K$, i.e. the ring of integers of $K$. The goal of this talk is to present a direct construction of the categorical equivalence between Wach modules over $\mathbf{A}_K^+$ and prismatic $F$-crystals over $O_K$. If time permits, we will also mention generalisation of our construction to the relative case as well as relationships between relative Wach modules, $q$-connections and filtered $(\varphi, \partial)$-modules.