Let \(U\) be a universe. Then étale morphisms in \(\mathord {\textnormal {\textsf {Topos}}}(U)\) are closed under composition.
Proof
Let \(X\) be a \(U\)-topos, let \(a:\mathord {\textnormal {\textsf {Sh}}}(X)\) be an object, and let \(s:\mathord {\textnormal {\textsf {Sh}}}(X)/a\) be an object. Let \(u:b\rightarrow a\) denote the morphism that represents \(s\). We have the equivalence \((\mathord {\textnormal {\textsf {Sh}}}(X)/a)/s\simeq \mathord {\textnormal {\textsf {Sh}}}(X)/b\). Then [000P] applies.