One knows by standard Algebraic Geometry that for any morphism $f:X \rightarrow Y$ of schemes one has canonical bijections

$$\operatorname{Hom}_X(f*G,F)\simeq \operatorname{Hom}_Y(G,f_{*}F).$$

Question: is it right that this map sends isomorphisms to isomorphisms?

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What are $F, G$ here? (From the answer I guess that they are sheaves, but it's worth clarifying.) –  Qiaochu Yuan Oct 25 '11 at 4:07

No. Take a closed immersion $f$ and $G=O_Y$, $F=O_X$. Then $f^*G=F$, but $G$ can't be isomorphic to $f_*F$ if $f$ is not an isomorphism.