Let $T, P$ be two topoi, and $f:T \longrightarrow P$.


Does there exist two site $S\_{T}, S\_{P}$ and a morphism $g: S\_{T} \longrightarrow S\_{P}$ such that $f$ is induced by $g$ ? 



question of topos and site
Yes: every morphism of Grothendieck toposes arises as a morphism of sites. (However, the site may depend on the morphism.) This is Corollary C2.3.10 in Johnstone's *Sketches of an elephant*. The argument goes something like this: pick any site $\mathcal{S}\_\mathcal{P}$ for the codomain $\mathcal{P}$; then we may take as $\mathcal{S}\_\mathcal{T}$ some subcategory of $\mathcal{T}$ that contains a site for $\mathcal{T}$ as well as the images under $f^\*$ of all representable sheaves on $\mathcal{S}\_\mathcal{P}$.


