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 $g$ ?
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 $g$ ?
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 $\mathcal{S}_\mathcal{P}$.