The Stacks project

Proof. If $f$ is étale, then $g$ is étale by Lemma 88.23.2. Conversely, assume $g$ is étale. Since $f$ is an isomorphism over $U$ we see that $f$ is étale over $U$. Thus it suffices to prove that $f$ is étale at any point of $X'$ lying over $T$. Denote $Z \subset X$ the reduced closed subspace whose underlying topological space is $|Z| = T \subset |X|$, see Properties of Spaces, Definition 66.12.5. Letting $Z_ n \subset X$ be the $n$th infinitesimal neighbourhood we have $X_{/T} = \mathop{\mathrm{colim}}\nolimits Z_ n$. Since $X'_{/T} = W \to X_{/T}$ we conclude that $f^{-1}(Z_ n) = X' \times _ X Z_ n \to Z_ n$ is étale by the assumed étaleness of $g$. By More on Morphisms of Spaces, Lemma 76.20.3 we conclude that $f$ is étale at points lying over $T$. $\square$