Remark 6.19.
The left adjoint of the restriction of \(u^*\) is given by the composite
\[\Shv_{\tau}(C) \hookrightarrow \PSh(C) \xrightarrow{u_!} \PSh(D) \xrightarrow{L_{\tau'}} \Shv_{\tau'}(D),\]
where \(u_!\) is left Kan extension along \(u\). In particular, any continuous functor \(u\) for which the left Kan extension functor \(u_!\) is left exact is a continuous morphism of sites.
Note that \(u_!\) is left exact whenever \(u\) admits a left adjoint, since then \(u_!\) is given by restriction along that left adjoint.