Let \(U\) be a universe. Then a morphism \(f:Y\rightarrow X\) in \(\mathord {\textnormal {\textsf {Topos}}}(U)\) is étale if and only if the morphism of lex \(U\)-cocomplete categories \({f}^{*}:\mathord {\textnormal {\textsf {Sh}}}(X)\rightarrow \mathord {\textnormal {\textsf {Sh}}}(Y)\) is étale.