Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 1 / d / Solution

The tangent-normal short exact sequencegives $\det (TX{|}_Y)\cong \det (TY)\otimes N_{Y/X}$. Dualizing and using the canonical bundle,Since $N_{Y/X}\cong \mathcal{O}(Y){|}_Y$, rearrangement proves the Adjunction formula