Synthetic topos theory
[000Q] Proposition

Let \(U\) be a universe. Then all the identity morphisms in \(\mathord {\textnormal {\textsf {Topos}}}(U)\) are étale.

Proof

Let \(X\) be a \(U\)-topos. We have the equivalence \(\mathord {\textnormal {\textsf {Sh}}}(X)\simeq \mathord {\textnormal {\textsf {Sh}}}(X)/\mathord {\textnormal {\textsf {1}}}\). Then [000P] applies.