Let $X$ be a complex manifold, $Y$ a sub-manifold, and $i \colon Y \to X$ the corresponding embedding. Then one can prove that the corresponding canonical bundles satisfy $$ \omega_X \big|_Y = \omega_Y \otimes \det(N_{Y/X})^* \label{eq:adjunction} \tag{1} $$ where $\omega_X \big|_Y = i^*\omega_X$ and $N_{Y/X} = TX \big|_Y \big/ TY$ is the normal bundle.
Why is this called the adjunction formula? I suppose that this is a special case of some adjunction, but I can't figure out the corresponding functors.
I thought that maybe this relates to the identity for the $\mathcal{Hom}$ bundle $$ \mathcal{Hom}(E,F) = E^* \otimes F $$ which applied to \ref{eq:adjunction} gives $$ \mathcal{Hom}(\det(N_{Y/X}),\omega_Y) = \omega_X \big|_Y = \det(T^*X) \big|_Y = \mathcal{Hom}(\det(TX) \big|_Y, X \times \Bbb{C}) $$ but I still can't make out the functors.