What's the easiest example of a morphism of topoi that is not from that of a site? A topos is defined to be a category that's equivalent to the category of sheaves on a site. Morphisms between topoi is defined by a pair of adjoint functors that behave like pull-back/push-forward of sheaves. But I was told one of the cool thing about topos is that sometimes there are morphisms of topos that are not from morphisms of a site. When people talk about this they mention the word "crystalline"...
But is there a toy example I can play around with? What's the easiest example of this?
 A: As for you word "crystalline" it comes from the Crystalline Topos. In the book "Notes on crystalline cohomology" by Berthelot and Ogus, they show that a morphism $X\to X'$ of schemes (over a fixed base $S$) induces a morphism of the associated crystalline topoi althugh there is no morphism of the corresponding sites. This is discussed in Section 5, page 5.1.
A: Let $X$ be a scheme. Let $S$ be the site of open subschemes with the Zariski topology, and let $S'$ be the site of open affine subschemes with the Zariski topology. Let $T$ and $T'$ be the associated toposes. Let $f\colon T\to T'$ be the topos map where $f^*(U)=U$ for any affine open subscheme $U$. Then $f$ is an equivalence because open affines form a base for the topology. Let $g$ be its inverse. Then $g^*$ does not restrict to a map of sites: If $V$ is an open subscheme, then $g^*(V)$ is the sheaf "represented by $V$" (i.e. it sends an affine open $U$ to $\mathrm{Hom}_X(U,V)$), but if $V$ is not affine, then $g^*(V)$ is not represented by an object in $S'$.
