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?