Let $X$ be a topological space and $\varphi \colon \mathcal{F} \to \mathcal{G}$ be a sheaf morphism on $X$.I know that surjectiveness of $\varphi$ means surjective stalk homomorphisms $\varphi_x \colon \mathcal{f}_x \to \mathcal{G}_x $ for any $x \in X$, but the section maps $\varphi_{U} \colon \mathcal{F}(U) \to \mathcal{G}(U)$ need not be surjective. I already know some example of surjective $\varphi \colon \mathcal{F} \to \mathcal{G}$ but not surjective $\varphi_{U} \colon \mathcal{F}(U) \to \mathcal{G}(U) $.
Now I’m studying a definition of spin structure on differentiable manifold.It appears surjective sheaf morphisms $\exp \colon \underline{\mathbb{R}} \to \underline{U(1)}$and $Ad \colon \underline{Spin(n)} \to \underline{SO(n)} $.Where, they are sheaf morphisms between smooth functions and 1-dimensional unitary groups, n-dimensional spin groups and special orthonormal groups on differential manifold.
Question. Are these sheaf morphisms always induce surjections on sections?I think it’s not true. If it’s not true, please construct some example.