Is $\mathbb{Q}$-factoriality preserved under contraction?

Question

Let ($X$,$\Delta$) be projective klt pair and $f \colon X \rightarrow Z$ be contraction of ($K_X + \Delta$) - negative extremal ray $R$. If $X$ is $\mathbb{Q}$ -factorial and $\mathrm{dim}Z < \mathrm{dim} X $, then Z is $\mathbb{Q}$- factorial?

I have read proof of above claim in Kollar and Mori "Birational geometry of algebraic varieties" but I have some trouble.The proof is following:

Let $D$ be effective weil divisor on $Z$ and $Z_0 \subset Z$ be smooth locus of $Z$. $D'$ be closure of $f^{-1}(D \cap Z_0 )$. By $\mathbb{Q}$-factoriality of X, $mD'$ is cartier for some $m \in \mathbb{Z}$. Since $D'$ does not meet general fibre of $f$ we have $D'.R = 0$ Hence $mD'$ is pullback of some cartier divisor on $Z$ and $D$ is $\mathbb{Q}$ cartier.

My trouble is the last statement.

Why $D$ is $\mathbb{Q}$ cartier since $mD'$ is pullback of some cartier divisor on Z?

Answer 1

Let $D''$ be a Cartier divisor on $Z$ such that $f^*\mathscr O_Z(D'')\simeq \mathscr O_X(mD')$. Then by the projection formula $\mathscr O_Z(D'')\simeq f_*\mathscr O_X(mD')$ (since $f_*\mathscr O_X\simeq \mathscr O_Z$).

On the other hand, from the construction we see that $\mathscr O_Z(D'')|_{Z_0}\simeq f_*\mathscr O_X(mD')|_{Z_0}\simeq \mathscr O_Z(mD)|_{Z_0}$. Finally, because $Z$ is normal, the complement of $Z_0$ has at least codimension $2$ in $Z$. Both $\mathscr O_Z(D'')$ and $\mathscr O_Z(mD)$ are reflexive sheaves of rank $1$, and they are isomorphic on the complement of a codimension $2$ closed set, so they are isomorphic everywhere. Hence $mD\sim D''$ which is Cartier.