Let $f: X \to Y$ be a morphism of schemes and $\mathcal{M}, \mathcal{N}$ be $\mathcal{O}_Y$-modules. Suppose $f^*\mathcal{M} \cong f^* \mathcal{N}$. Under which assumptions can I conclude that already $\mathcal{M} \cong \mathcal{N}$?
This is clearly wrong in the general as one can see by $f$ being the inclusion of a point, so surjectivity/dominance of $f$ will be one of these conditions, but is this condition already enough? I was playing around with the adjunction of pushforward and pullback but currently I am stuck.
I am mainly interested in the case of $\mathcal{M}, \mathcal{N}$ being line bundles on a nodal curve $C$ over an algebraically closed field and $f$ being the normalisation of $C$, so I would be happy about a positive result in this case, but I think the general question is also educational.