Lemma 6.21.8. Let $f : X \to Y$ be a continuous map. Let $\mathcal{F}$ be a sheaf of sets on $X$ and let $\mathcal{G}$ be a sheaf of sets on $Y$. There are canonical bijections between the following three sets:

The set of maps $\mathcal{G} \to f_*\mathcal{F}$.

The set of maps $f^{-1}\mathcal{G} \to \mathcal{F}$.

The set of $f$-maps $\xi : \mathcal{G} \to \mathcal{F}$.

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