Consider a curve $C$. A morphism $f: C \to \mathbb{P}^1$ induces the dualizing sheaf $\omega_f$ endowed with following characteristic property: $$ f_*(\omega_f) = \underline {Hom} _{\mathcal{O}_{\mathbb{P}^1}}(f_*(\mathcal{O}_X), \omega_{\mathbb{P}^1})$$
where $\omega_{\mathbb{P}^1} = \mathcal{O}_{\mathbb{P}^1}(-2)$.
My question refers to my former question Dualizing Sheaf unique determined concerning the uniqueness of the dualizing sheaf.
I want to know if there exist a bijective correspondence between
quasicoherent $f_*(\mathcal{O}_C)$ modules $\mathcal{M}$
and
quasicoherent $\mathcal{O}_C$ modules $\mathcal{F}$ with property: $\mathcal{M}= f_*(\mathcal{F})$
If yes, does anybody have a good source/reference where this correspondence is introduced/explained. Futhermore does this correspondence has a name?