Injectives obstruct Fourier-Mukai functors

We use injectives as a big tilting object to obstruct liftability of exact functors to the $\text{dg}$-level. We use the inclusion of injectives into the canonical heart as a replacement for tilting objects in computations of the characteristic morphism. Then we apply this construction to proofs of non-liftability of candidate non-Fourier-Mukai functors, i.e.\ functors that do not admit an $\text{Ainfty}$/$\text{dg}$-lift. This approach allows explicit computation of the obstruction against an $\text{Ainfty}$-lift. We in particular observe that this computation gives for smooth degree $d>2$ hypersurfaces an abundance of non-Fourier-Mukai functors.