Projection formula for flat morphisms Assume we have a flat morphism $f: X\rightarrow Y$ of schemes
such that $f_{*}O_X=O_Y$. Given a locally free sheaf $E$ of finite rank, the projection formula gives $f_{*}f^{*}E=f_{*}(O_X\otimes f^{*}E)=f_{*}O_X\otimes E=E$.
Using the flatness of $f$, can we replace locally free of finite rank by coherent or quasi coherent? 
This is indicated in this answer, but I don't see how to use the flatness of $f$ to prove such a result.
Any advice on how to see this is appreciated. Maybe this can be translated in a commutative algebra question which is easier? References are also welcome.

For any (quasi-) coherent sheaf $F$ on $Y$and any $f:X\to Y$, you have a natural map $F\to f_*(f^*F)$ and so to check this is an isomorphism is local on $Y$. So, if $Y$ is locally Noetherian, we are reduced to the situation for a coherent sheaf $F$ with a presentation $\mathcal{O}_Y^m\to\mathcal{O}_Y^n\to F\to 0$ and now use $f_*\mathcal{O}_X=\mathcal{O}_Y$ to finish the proof.
