The Adjunction Formula is given to be
$K_V = (K_X \otimes [V])_V$
Where $K_V$ is the canonical class on $V\subset X$ and $K_X$ of $X$. And $[V]$ denotes the line bundle associated to $V$.
Now say, Instead of the canonical bundle on $V$, that I'm interested in some general Line bundle $[D]$. Is there some equivalent formula like
$[D]_V = ([D]_X \otimes [V])_V$?
Or how do you actually restrict a line bundle on $X$ to $V$? How do their chern classes relate?