Let $X \subset \mathbb{P}^5$ be a Pfaffian cubic fourfold (or one of the other known rational cubic fourfolds). It is known by Kuznetsov's Homological Projective Duality that $\mathcal{K}u(X) \simeq D^b(S)$ where $\mathcal{K}u(X)$ is the Kuznetsov component of $D^b(X)$, and $S$ is a K3 surface.

What is the explicit definition of the functor $\Phi \colon \mathcal{K}u(X) \xrightarrow{\sim} D^b(S)$?