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 $\Phi \colon \mathcal{K}u(X) \xrightarrow{\sim} D^b(S)$?
It depends on each case. In general there is an explicit birational map from $X$ to a rational variety (typically $P^4$, or a quadric, as it is the case for pfaffian cubics), whose indeterminacy locus is (a birational model of) the K3. Then you just follow the factorization of the birational map and get your function. For instance a pfaffian cubic 4fold contains a quartic scroll $S$ and the linear system of quadrics through $S$ sends $X$ to a smooth quadric 4fold. If I remember properly, the bisecant lines to $S$ contained in $X$ are contracted to the image in $P^5$ of a degree 14 K3 blown up in a point (which is contained in the quadric). So the inverse map (given by cubics) blows up the K3 and you find its derived category where it should be.
