Let $f: (X, O_X) \rightarrow (Y, O_Y)$ be a scheme morphism, $F$ - module over $O_X$, $G$ - module over $O_Y$. How to prove, that $ Hom_{O_X}(f^*G, F) = Hom_{O_Y}(G, f_* F). $ Please give the most detailed proof.
Functors $f^*$ and $f_*$ on the category of sheafs of modules (Hartshorne)
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algebraic-geometry
category-theory
sheaf-theory
adjoint-functors
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1Note that, if $X$ and $Y$ are affine, this follows from the tensor-hom adjunction. So it is sufficient to reduce to affine patches. – 2016-01-17