\mathcal O_Y(lH^\dag))\otimes 
\mathcal O_Y\to 
f_*\omega^{\otimes k}_{X/Y}\otimes 
\omega_Y\otimes \mathcal O_Y(lH^\dag)$$ is surjective on $U$ for all integers $k\geq 1$ and $l\geq \frac{n(n+1)}{2}+1$.

Theorem \[z-thm1.5\] is a special case of Theorems \[y-thm1.7\] and \[y-thm1.8\]. In this paper, we will establish Theorems \[y-thm1.7\] and \[y-thm1.8\] by our new theory of mixed-$\omega$-sheaves.

\[y-thm1.7\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$ with $\dim Y=n$. Let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier and that $(X, \Delta)$ is log canonical over a nonempty Zariski open set of $Y$. Let $L$ be a Cartier divisor on $X$ with $L\sim _{\mathbb R}k(K_{X/Y}+\Delta)$ for some positive integer $k$. Let $H$ be a big Cartier divisor on $Y$ such that $|H|$ is free. Then $$f_*\mathcal O_X(L)\otimes \mathcal O_Y(K_Y+lH)$$ is generically generated by global sections for every $l\geq n+1$. Moreover, $$\left(\bigotimes ^s f_*\mathcal O_X(L)\right)^{**} 
\otimes \mathcal O_Y(K_Y+lH)$$ is generically generated by global sections for all integers $s\geq 1$ and $l\geq n+1$.

\[y-thm1.8\] In Theorem \[y-thm1.7\], we assume that $H^\dag$ is a nef and big Cartier divisor on $Y$ such that $|H^\dag|$ is not necessarily free. Then we have that $$f_*\mathcal O_X(L)\otimes \mathcal 