Definition 6.23. (Cocontinuous morphism)
Let \((C,\tau)\) and \((D,\tau')\) be Grothendieck sites. A functor \(u\colon C \to D\) is called a cocontinuous morphism of sites if the right Kan extension functor restricts to a functor
\[u_*\colon \Shv_{\tau}(C) \to \Shv_{\tau'}(D).\]
In particular, \(u_*\) is a morphism of topoi, with left exact left adjoint given by the composite
\[\Shv_{\tau'}(D) \hookrightarrow \PSh(D) \xrightarrow{u^*} \PSh(C) \xrightarrow{L_{\tau}} \Shv_{\tau}(C).\]
If we denote by \(\Site^{\cocont}\) the category of Grothendieck sites and cocontinuous morphisms, then the assignment \((C,\tau) \mapsto \Shv_{\tau}(C)\) defines a functor
\[\Site^{\cocont} \to \Topos,\]
obtained by restricting the right Kan extension functoriality of the presheaf construction \(\PSh\colon \Cat \to \PrR\).