Construction 7.14.
Let \((C,C^{\ad},\tau)\) be a geometric site. The category \(C^{\ad}\) inherits a topology \(\tau^{\ad}\): a sieve in \(C^{\ad}_{/X}\) is covering if its image generates a \(\tau\)-covering sieve in \(C_{/X}\). Restriction along \(j\colon C^{\ad}\hookrightarrow C\) defines a functor
\[j^*\colon \Shv_{\tau}(C)\longrightarrow \Shv_{\tau^{\ad}}(C^{\ad}).\]
Its left adjoint \(j_!\) is obtained by left Kan extension followed by sheafification.