# A slightly canonical way to associate a scheme to a Noetherian spectral space

Let $$C$$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $$\mathrm{AffSch}$$ be the category of Noetherian affine schemes (morphisms are morphisms of schemes). Let $$\mathrm{SpecSpc}$$ be the category of Noetherian spectral topological spaces (morphisms are continuous maps).

We have an obvious functor $$I:C\rightarrow \mathrm{SpecSpc}$$ and the forgetful functor $$F:\mathrm{AffSch}\rightarrow \mathrm{SpecSpc}$$. Does there exist a functor $$G:C\rightarrow \mathrm{AffSch}$$ such that $$F\circ G=I$$?

A slightly less functorial question: is every Noetherian spectral space the underlying space of a Noetherian scheme?