New examples of non-Fourier-Mukai exact functors via non-isomorphic octahedra

We study a triangulated category $\mathscr S$ that admits a full and strong exceptional sequence of three objects with one-dimensional Hom spaces. We show that the isomorphism classes of exact functors from $\mathscr S$ to another triangulated category $\mathscr T$ are in bijection with the isomorphism classes of octahedra in $\mathscr T$ satisfying a natural condition. As an application, we construct an exact functor from $\mathscr S$ to $\mathbf D^b(\Bbbk[x]\text{-}\mathrm{mod})$ that does not admit a dg lift. This provides an explicit example of a non-Fourier-Mukai exact functor between $\mathbf D^b(\mathbb P^2)$ and $\mathbf D^b(\mathbb P^1)$.