Conormal line of a smooth analytic hypersurface ($[-Y]{|}_Y\cong N_{Y/X}^{\ast }$)

For reduced local equations $f_j=u_{ji}{f}_i$ with $u_{ji}$ a holomorphic unit, the holomorphic line bundle associated to a divisor $[-Y]$ has local generators $f_i$ and transition $u_{ji}$. Differentiation along $Y$ gives $d{f}_j{|}_Y=u_{ji}{|}_{Y}d{f}_i{|}_Y$. Sending the restricted local generator to $d{f}_i{|}_Y$ gives the conormal isomorphism. The term $f_{i}d{u}_{ji}$ disappears only after restricting to $Y$.