Is the invertible sheaf associated to the pullback of a Cartier divisor the pullback of the invertible sheaf associated to that divisor?

I'm trying to figure out some facts about the pullback of Cartier divisors, but honestly I'm having a hard time describing the pullback of the sheaf associated to divisor.

Let $\phi\colon X \to Y$ be a dominant morphism of schemes. Suppose $X, Y$ are both integral, noetherian, separated. If $\xi_X$ and $\xi_Y$ are the generic points of $X$ and $Y$, the dominance hypothesis says that $\phi(\xi_X) = \xi_Y$, so $\phi^\#$ maps rational functions on $Y$ to rational functions on $X$. Let $D = (U_i, f_i)$ be a Cartier divisor on $Y$, and suppose that $\phi$ is such that $ D' = (\phi^{-1}(U_i), \phi^\#(f_i))$ is a Cartier divisor on $X$ (is this always the case under these hypotheses?)

Do the sheaves associated to $D$ and $D'$ satisfy $\phi^*(O_Y(D)) = O_X(D')$? If D is effective and associated to a closed subscheme $Z$ of $Y$, is $D'$ effective and associated to the closed subscheme $\phi^{-1}(Z)$?