Let $(X, \mathcal{O}_X)$ be a complex manifold, which we can take to be projective. A coherent subsheaf $\mathscr{F}$ of some sheaf $\mathscr{G}$ is said to be saturated in $\mathscr{G}$ if the quotient sheaf $\mathscr{G}/\mathscr{F}$ is torsion-free. Further, we can define the saturation of $\mathscr{F}$ inside $\mathscr{G}$ to be the kernel of the map $$\mathscr{G} \to (\mathscr{G}/\mathscr{F})/(\text{torsion}).$$
What is the intuition for the saturation of a subsheaf? Are there some elementary intuition-granting examples?
Example: Let $E$ be an elliptic curve and $C$ a hyper elliptic curve with an involution $\sigma$ such that the quotient $\hat{C} := C/\sigma$ is $\mathbb{P}^1$. Take $\tau$ to be a translation of order $2$ on $E$ and define $X = (C \times E)/(\sigma, \tau)$. Let $f : X \to C'$ be the Iitaka fibration of $X$, and let $\mathscr{L} = f^{\ast}K_{C'}$, which is a subsheaf of $\Omega_X^1$. The saturation is then the kernel of $$\Omega_X^1 \to (\Omega_X^1 / \mathscr{L}) / \text{(torsion)}.$$
Can we get an explicit description of the saturation in this case?
