The Stacks project

Lemma 6.30.14. Let $f : X \to Y$ be a continuous map of topological spaces. Let $(\mathcal{C}, F)$ be a type of algebraic structures. Let $\mathcal{F}$ be a sheaf with values in $\mathcal{C}$ on $X$. Let $\mathcal{G}$ be a sheaf with values in $\mathcal{C}$ on $Y$. Let $\mathcal{B}$ be a basis for the topology on $Y$. Suppose given for every $V \in \mathcal{B}$ a morphism

\[ \varphi _ V : \mathcal{G}(V) \longrightarrow \mathcal{F}(f^{-1}V) \]

of $\mathcal{C}$ compatible with restriction mappings. Then there is a unique $f$-map (see Definition 6.21.7 and discussion of $f$-maps in Section 6.23) $\varphi : \mathcal{G} \to \mathcal{F}$ recovering $\varphi _ V$ for $V \in \mathcal{B}$.

Proof. This is trivial because the collection of maps amounts to a morphism between the restrictions of $\mathcal{G}$ and $f_*\mathcal{F}$ to $\mathcal{B}$. By Lemma 6.30.10 this is the same as giving a morphism from $\mathcal{G}$ to $f_*\mathcal{F}$, which by Lemma 6.21.8 is the same as an $f$-map. See also Lemma 6.23.1 and the discussion preceding it for how to deal with the case of sheaves of algebraic structures. $\square$

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