Definition 7.41.
A map \(f\colon A \to B\) of analytic rings is said to be:
Proper if the functor \(f_*\) induces an equivalence
\[\Mod_B \iso \Mod_{B^{\triangleright}} \times_{\Mod_{A^{\triangleright}}} \Mod_A.\]An open immersion if the functor \(f^*\) admits a further left adjoint
\[f_!\colon \Mod_B \to \Mod_A\]which is fully faithful and \(\Mod_A\)-linear. Equivalently, it is fully faithful and satisfies the projection formula
\[f_!(f^*(M) \otimes_B N) \cong M \otimes_A f_!(N).\]\(!\)-able if the canonical map of analytic rings
\[(B^{\triangleright}, \Mod_{B^{\triangleright}} \times_{\Mod_{A^{\triangleright}}} \Mod_A) \longrightarrow (B,\Mod_B)\]is an open immersion.