  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
$\mathcal{A}_\infty$/$\mathrm{dg}$-lift. This approach allows explicit
computation of the obstruction against an $\mathcal{A}_\infty$-lift. We in
particular observe that this computation gives for smooth degree $d>2$
hypersurfaces an abundance of non-Fourier-Mukai functors.
