It's relatively simple to show that the geometric morphisms $ \mathbf{Set} \to \mathrm{Sh}(\mathbf{CRing}^\mathrm{op}_{\mathrm{fp}}, \mathrm{Zar})$ correspond to local rings.

More precisely, since the site has finite limits, the inverse image functors are induced by functors $\mathbf{CRing}^\mathrm{op}_{\mathrm{fp}} \to \mathbf{Set}$ that preserve finite limits and map covers to epimorphic families, and these are precisely the functors (isomorphic to) $\mathbf{CRing}(-, \mathcal{O})$ for local rings $\mathcal{O}$.

But we often consider Zariski sheaves on larger sites. For example, one might assume a small inaccessible cardinal $\kappa$ and consider the site $(\mathbf{CRing}^\mathrm{op}_{\kappa}, \mathrm{Zar})$ of *all* $\kappa$-small affine schemes.

The functors $\mathbf{CRing}(-, \mathcal{O})$ still define (inverse image parts of) geometric points, and every point whose inverse image part preserves *all* limits of affine schemes must be of this form.

But in general, the inverse image part of a geometric point is only required to preserve finite limits, so a priori there might be additional points.

Is there an argument that every geometric point of such a site is given as above? Or can there be points not of that form?