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

We study a triangulated category $\mathcal{S}$ 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 $\mathcal{S}$ to another triangulated category $\mathcal{T}$ are in bijection with the isomorphism classes of octahedra in $\mathcal{T}$ satisfying a natural condition. As an application, we construct an exact functor from $\mathcal{S}$ to ${\mathbf{D}}^b(\mathbb{k}[x]\text{-}\mathrm{m}\mathrm{o}\mathrm{d})$ 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 ${\mathbf{D}}^b({\mathbb{P}}^1)$.