Synthetic topos theory
[0013] Proposition

Let \(U\) be a universe. Then, for any \(U\)-topos \(X\), the category \(\mathord {\textnormal {\textsf {Etale}}}(X)\) has \(U\)-small colimits. Moreover, for any morphism \(f:X\rightarrow Y\) in \(\mathord {\textnormal {\textsf {Topos}}}(U)\), the pullback functor \({f}^{*}:\mathord {\textnormal {\textsf {Etale}}}(Y)\rightarrow \mathord {\textnormal {\textsf {Etale}}}(X)\) commutes with \(U\)-small colimits.

Proof

By [0012].