  We show that the adjunction counits of a Fourier-Mukai transform $\Phi$ from
$D(X_1)$ to $D(X_2)$ arise from maps of the kernels of the corresponding
Fourier-Mukai transforms. In a very general setting of proper separable schemes
of finite type over a field we write down these maps of kernels explicitly --
facilitating the computation of the twist (the cone of an adjunction counit) of
$\Phi$. We also give another description of these maps, better suited to
computing cones if the kernel of $\Phi$ is a pushforward from a closed
subscheme $Z$ of $X_1 \times X_2$. Moreover, we show that we can replace the
condition of properness of the ambient spaces $X_1$ and $X_2$ by that of $Z$
being proper over them and still have this description apply as is. This can be
used, for instance, to compute spherical twists on non-proper varieties
directly and in full generality.
