Definition 6.18. (Continuous morphism)
Let \((C,\tau)\) and \((D,\tau')\) be Grothendieck sites. A functor \(u\colon C \to D\) is called continuous if the restriction functor \(u^*\colon \PSh(D) \to \PSh(C)\) preserves sheaves, i.e. restricts to a functor
\[u^*\colon \Shv_{\tau'}(D) \to \Shv_{\tau}(C).\]
We say that \(u\) is a continuous morphism of sites if this restriction is a morphism of topoi, i.e. its left adjoint is left exact. We denote the category of Grothendieck sites and continuous morphisms of sites by \(\Site^{\cont}\). By definition, the assignment \((C,\tau) \mapsto \Shv_{\tau}(C) \subseteq \PSh(C)\) determines a contravariant functor
\[\Shv(-)\colon (\Site^{\cont})\catop \to \Topos,\]
obtained by restricting the precomposition functoriality of the presheaf construction \(\PSh(-)\colon \Cat\catop \to \PrL\).