Let MAN be the category of differentiable manifolds with smooth maps as morphisms. Let LRS be the category of locally ringed spaces. Now I know that there exists a functor $\textbf{MAN}\to\textbf{LRS}$ that associates $$ M\mapsto(M, C^\infty) $$
where $ C^\infty$ is the sheaf of smooth functions on $M$. My question is whether this functor is fully faithfull, that is, given a morphism of locally ringed spaces $$ (M, C^\infty)\to(N, C^\infty) $$ then is the associated function $M\to N$ smooth? I know that we will have to use that the induced map at stalks is local at some point, but I can't see how.
