The Stacks project

Lemma 6.23.1. Let $f : X \to Y$ be a continuous map of topological spaces. Suppose given sheaves of algebraic structures $\mathcal{F}$ on $X$, $\mathcal{G}$ on $Y$. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be an $f$-map of underlying sheaves of sets. If for every $V \subset Y$ open the map of sets $\varphi _ V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}V)$ is the effect of a morphism in $\mathcal{C}$ on underlying sets, then $\varphi$ comes from a unique $f$-morphism between sheaves of algebraic structures.