On adjunctions for Fourier-Mukai transforms

We show that the adjunction counits of a Fourier-Mukai transform $\mathrm{\Phi }$ 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 $\mathrm{\Phi }$. We also give another description of these maps, better suited to computing cones if the kernel of $\mathrm{\Phi }$ 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.