Guitart states in "[Toute theorie est algebrique et topologique][1]" (Proposition 17) without proof (at least, I don't understand the hints there) that the category $\mathbf{Sch}$ of schemes is the category of models of a large mixed [sketch][2]. Presumably, it is built using [Burroni's sketch][3] of $\mathbf{Top}$.

**Questions.**
1) Can someone explain more how that sketch for $\mathbf{Sch}$ looks like?
2) Perhaps a first step would be to find sketches for the categories $\mathbf{RS}$ and $ \mathbf{LRS}$ of (locally) ringed spaces, but do they exist? I suspect that for this we first need to find a categorical characterization of local homeomorphisms ($\equiv$ sheaves) internal to $\mathbf{Top}$?
3) Is there a large *limit* sketch for the category of affine schemes, i.e. $\mathbf{CRing}^{\mathrm{op}}$? That would be a huge surprise. In case it helps, $\mathbf{CRing}^{\mathrm{op}}$ is not Isbell-compact. Is there, at least, a large mixed sketch for affine schemes?


  [1]: https://eudml.org/doc/91733
  [2]: https://ncatlab.org/nlab/show/sketch
  [3]: https://mathoverflow.net/questions/277329/what-was-burronis-sketch-for-topological-spaces?noredirect=1&lq=1