The Stacks project

Proof. The statement makes sense by Lemma 88.17.1. To see that it is true assume we have morphisms $B \to A$ and $B \to C$ in $\textit{WAdm}^{Noeth}$ and that $B \to A$ is rig-smooth and $B \to C$ is adic (Formal Spaces, Definition 87.6.1). Then we can choose an ideal of definition $I \subset B$ such that the topology on $A$ and $C$ is the $I$-adic topology. In this situation it follows immediately that $A \widehat{\otimes }_ B C$ is rig-smooth over $(C, IC)$ by Lemma 88.4.5. $\square$