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Let $f : X \rightarrow Y$ be a continuous map. Let $\Im$ be a sheaf on $Y$ and $U \subseteq Y$ be an open subset. I need to show that there is a bijective correspondence between $(f^{-1} \Im)(f^{-1}(U))$ and $(f_{*}f^{-1}\Im)(U)$.

Here, $f^{-1} \Im$ is the inverse image of $\Im$ and $f_{*} \Im$ is the direct image of $\Im$.

Now since $f_{*} \Im(U) := \Im(f^{-1}(U))$, so $$ (f^{-1} \Im)(f^{-1}(U)) = f^{-1} f_* \Im(U) $$

But I'm not able to make any progress. Any help with this!

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    Are you confusing the sheaf on $X$ with the sheaf on $Y$ in your progress part? – 2017-01-25

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Given a continuous function $f:X\to Y$ and a sheaf $F$ on $X$, by definition the direct image sheaf $f_*F$ is given by $f_*F(U) : = F(f^{-1}U)$. Now, the inverse image sheaf $f^{-1}\mathfrak{J}$ where $\mathfrak{J}$ is a sheaf on $Y$, is a sheaf on $X$. Hence by definition we have that $$f_*f^{-1}\mathfrak{J}(U)= f^{-1}\mathfrak{J}(f^{-1}U). $$

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    Just to make sure - $U$ is an open subset of $Y$. right? – 2017-01-26
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    Yes $U$ is an open subset of $Y$. – 2017-01-26