The Stacks project

Proof. Let $J \subset B$ be an open ideal with inverse image $I \subset A$. Then the closure of $JC$ contains the closure of $IC$. Hence this closure is open as $A \to C$ is taut. Let $I_\lambda$ be a fundamental system of open ideals of $A$. Let $K_\lambda$ be the closure of $I_\lambda C$. Since $A \to C$ is taut, these form a fundamental system of open ideals of $C$. Denote $J_\lambda \subset B$ the inverse image of $K_\lambda$. Then the closure of $J_\lambda C$ is $K_\lambda$. Hence we see that the closures of the ideals $JC$, where $J$ runs over the open ideals of $B$ form a fundamental system of open ideals of $C$. $\square$