The stacks project says that a smooth map $f:M\rightarrow N$ naturally gives rise to an arrow between the sheaves of smooth functions. I don't understand how this fits in with the notion of a local diffeomorphism: a local diffeo is a smooth map whose germs are diffeomorphisms. If every such arrow induced (and was induced from) an arrow of sheaves, then every local diffeo would be a global diffeo.
What am I missing?