Let $f:X\rightarrow Y$ be a morphism of schemes and let $\mathcal{F}$ be a sheaf on $Y$. Then there is a natural map $$\Psi:\mathcal{F} \rightarrow f\_{\ast}f^{\ast}(\mathcal{F})$$ and localizing $$\Psi\_p:\mathcal{F}\_p \rightarrow f\_{\ast}f^{\ast}(\mathcal{F})\_p=f\_{p,\ast}f\_p^{\ast}(\mathcal{F}\_p)$$


I'm interested in learning under what conditions on $X,Y,f,\mathcal{F},p$ is $\Psi\_p$ an isomorphism. E.g. is it an isomorphism if $\mathcal{F}$ is quasi-coherent and $f$ is étale?


Thanks in advance for any insight.



Sheaf isomorphism
If $f\_\*O\_X=O\_Y$ then this is true for any locally free (or flat) sheaf (and if $f$ is flat then for any quasicoherent sheaf) by projection formula. 


