, which we call a fundamental theorem of the theory of mixed-$\omega$-sheaves.  \[y-thm0.00\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$. Let $L$ be a Cartier divisor on $X$ and let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier. Let $D$ be an $\mathbb R$-divisor on $Y$. Let $k$ be a positive integer with $k\geq 0$. Assume the following conditions:  -   $(X, \Delta)$ is log canonical 