I'm reading the chapter on manifolds in Wedhorn's Manifolds, Sheaves, and Cohomology. At some point the author (in a somewhat terse manner) gives a characterization of the structure sheaf of a (pre)manifold (see Definition 1 below) in terms of the sheaf of real-valued locally ringed space morphisms. I'm a bit lost, so here's this question.
Let $ (X,\mathscr O_X) $ be a (locally) $ \mathbb R $-ringed space (meaning that $ \mathscr O_X $ is a sheaf of $ \mathbb R $-algebras on $ X $). Let $ \mathscr O_{X;\mathbb R} $ be the sheaf of real-valued (locally) ringed space morphism on $ (X,\mathscr O_X) $, i.e., let $ \mathscr O_{X;\mathbb R} $ be the sheaf on $ X $ that maps $$ \mathscr O_{X;\mathbb R}(U) = \{\text{$ (f,f^\flat)\colon (U,{\mathscr O_X}{\restriction_U})\to (\mathbb R,\mathscr C_{\mathbb R}^\infty) $ morphism of (L)RS}\} $$ for any $ U\subset X $ open, where $ \mathscr C_{\mathbb R}^\infty $ is the sheaf of smooth functions on the real line.
I'm trying to show that if $ (X,\mathscr O_X) $ satisfyies Definition 1 then the structure sheaf $ \mathscr O_X $ and $ \mathscr O_{X;\mathbb R} $ are isomorphic.
Definition 1. A locally ringed space $ (X,\mathscr O_X) $ is a premanifold if there exists an open cover $ (U_i)_{i\in I} $ of $ X $ such that for each $ i\in I $ $$ (U_i,{\mathscr O_X}{\restriction_{U_i}})\cong (\underline{U_i},\mathscr C_{\underline{U_i}}^\infty) $$ for some $ n\in \mathbb N $ and $ \underline{U_i}\subset \mathbb R^n $ open, where $ \mathscr C_{\underline{U_i}}^\infty $ is the sheaf of smooth real-valued functions on $ \underline{U_i} $.
Given $ U\subset X $ I'll try to define a morphism $$ i_U\colon \mathscr O_{X;\mathbb R}(U)\to \mathscr O_X(U) $$ by mapping a section $ (s,s^\flat) $ to the image $ s_{\mathbb R}^\flat(1_{\mathbb R})\in \mathscr O_X(U) $ (recall that $ s^\flat $ is a morphism of sheaves $ s^\flat\colon \mathscr C_{\mathbb R}^\infty\to s_*{\mathscr O_X}{\restriction_U} $, so that what I wrote type-checks).
So at this point I must show the following:
- that $ i_U $ is a homomorphism;
- that the $ i_U $s are the components of a morphism of sheaves;
- and that $ i $ is indeed an isomorphism.
I'm (somewhat) able to prove 2), but I easily get lost in the details when I try to show 1) and 3). Could you help me?