Corollary 4.40.
Consider morphisms of topoi \(T \xrightarrow{\phi} T' \xrightarrow{\psi} S\). If \(\psi\) and \(\psi\phi\) are étale morphisms, then so is \(\phi\).
Proof
Choose equivalences \(T\simeq S_{/X}\) and \(T'\simeq S_{/Y}\) identifying \(\psi\phi\) and \(\psi\) with the slice projections. By Proposition 4.38, the morphism \(\phi\) over \(S\) is classified by a morphism \(f\colon X\to Y\). Regarding \(f\) as an object of \(S_{/Y}\simeq T'\), there is an equivalence
\[T'_{/f}\simeq (S_{/Y})_{/f}\simeq S_{/X}\simeq T\]
under which the projection to \(T'\) identifies with \(\phi\). Hence \(\phi\) is étale.