If $f:X\rightarrow Y$ is a morphism of schemes and $M$ an $\mathcal{O}_Y$-module, then by definition $f^*M=f^{-1}M\otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X$, where $f^{-1}$ denotes the inverse image of sheaves. Is it true that $$f_*f^*M\cong f_*f^{-1}M\otimes_{f_*f^{-1}\mathcal{O}_Y} f_* \mathcal{O}_X$$ as sheaves? What if $f$ is surjective, or even faithfully flat?
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