In the paper ``Using the internal language of toposes in algebraic geometry'', Blechschmidt mentions several approaches to defining the big Zariski topos over a scheme. I have two questions, which are really only about working over $\mathrm{Spec}\mathbb{Z}$.
I have seen (for example on nLab) that $\mathrm{Sh}(\mathrm{CRing}_{fp}^{op})$ is the classifying topos of the theory of local rings. This is nice because then the points of the topos are exactly local rings in $\mathrm{Set}$. I was wondering what exactly does the restricting to finitely presented rings achieve? Is it necessary for proving it is the classifying topos or is it just to bypass size issues? If so...
Let $\mathscr{U}$ be a Grothendieck universe and define $\mathrm{Sh}(\mathrm{CRing}_{\mathscr{U}}^{op})$ as sheaves on the dual category of commutative rings in $\mathscr{U}$ with respect to the Zariski topology. Then is $\mathrm{Sh}(\mathrm{CRing}_{\mathscr{U}}^{op})$ the classifying topos of local rings in $\mathscr{U}$? If not, is it related in some way?
For reference, I am very much an amateur in topos theory so my questions may be ill-formulated. I apologise in advance.
