is obvious. So we assume that $s\geq 2$. We consider the following commutative diagram $$\xymatrix{
X^s_0 \ar[dr]^-{f^s}\ar[d]_-{p_s}\ar[r]^-q& X^{s-1}_0 \ar[d]^-{f^{s-1}}\\
X_0 \ar[r]_-f& Y_0
}$$ where $q=(p_1, \cdots, p_{s-1})$. Then we have $$\label{x-3.8}
\mathcal O_{X^s_0}\left(\sum _{i=1}^s p^*_i L_0\right)\simeq 
\mathcal O_{X^s_0}(p^*_sL_0)\otimes 
q^*\mathcal O_{X^{s-1}_0}\left(\sum _{i=1}^{s-1}p^*_iL_0\right).$$ Therefore, we obtain $$\begin{split}
f^s_*\mathcal O_{X^s_0}\left(\sum _{i=1}^s p^*_iL_0\right)
&\simeq  f_*{p_s}_*\left(\mathcal O_{X^s_0}(p^*_sL_0)\otimes 
q^*\mathcal O_{X^{s-1}_0}\left(\sum _{i=1}^{s-1}p^*_iL_0\right)\right)
\\
&\simeq 
f_*\left(\mathcal O_{X_0}(L_0)\otimes {p_s}_*q^*\mathcal O_{X^{s-1}_0}\left(
\sum _{i=1}^{s-1}p^*_iL_0\right)\right)
\\ 
&\simeq 
f_*\left(\mathcal O_{X_0}(L_0)\otimes f^*f^{s-1}_*\mathcal O_{X^{s-1}_0}\left(
\sum _{i=1}^{s-1}p^*_iL_0\right)\right)
\\ 
&\simeq 
f_*\left(\mathcal O_{X_0}(L_0)\otimes f^*\left(\bigotimes^{s-1}f_*\mathcal O_{X_0}(L_0)\right)\right) 
\\ 
&\simeq f_*\mathcal O_{X_0}(L_0)\otimes \bigotimes ^{s-1}f_*\mathcal O_{X_0}(L_0)
\\ &\simeq 
\bigotimes ^s f_*\mathcal O_{X_0}(L_0). 
\end{split}$$ Note that the first isomorphism follows from , the second one is due to the projection formula, the third one is obtained by the flat base change theorem, the fourth one is due to induction on $s$, and the fifth one follows from the projection formula. Anyway, we obtain the desired isomorphism.

Let us go back to the proof of Lemma \[x-lem3.6\]. We have an inclusion $$\begin{split}
\rho_*\mathcal O_{X^{(s)}_0}(L^{(s)}_0)&\subset 
\omega^{\otimes k}_{X^s_0/Y_0}\otimes \mathcal O_{X^s_0}\left(\sum 
_{i=1}^s p^*_i (L_0-kK_{X_0/Y_0})\right)\\ 
&\simeq \mathcal O_{X^s_0}\left(\sum_{i=1}^s p^*_i L_0\right)
\end{split}$$ by and Lemma \[x-rem3.5\], which is an isomorphism over a nonempty Zariski open set of $Y_0$. By taking $f^s_*$, we obtain a generically isomorphic injection $$\begin{split}
f^{(s)}_*\mathcal 