Maps of analytic rings

A map \(f\colon A \to B\) of analytic rings is a map \(A^{\triangleright} \to B^{\triangleright}\) satisfying the condition that the restriction-of-scalars functor \(\Mod_{B^{\triangleright}} \to \Mod_{A^{\triangleright}}\) restricts to a functor

We denote its left adjoint by

which is given by first including into \(\Mod_{A^{\triangleright}}\), then applying base change to \(\Mod_{B^{\triangleright}}\), and then reflecting back into \(\Mod_B\). We denote the resulting category of analytic rings by \(\AnRing\).