[001H] -- Synthetic topos theory

[001H] Proposition

Let $$U$$ be a universe and let $$V$$ be a universe greater than or equal to $$U$$ . Then the functor $C\mapsto \mathop {\Uparrow ^{V}}C:\mathord {\textnormal {\textsf {Logos}}}(U)\rightarrow \mathord {\textnormal {\textsf {Logos}}}(V)$ takes étale limits in $\mathord {\textnormal {\textsf {Logos}}}(U)$ to étale limits in $\mathord {\textnormal {\textsf {Logos}}}(V)$ .

Proof

Let $$C$$ be a $$U$$ -logos, let $$I$$ be a $$U$$ -small category, and let $A:I\rightarrow C$ be a functor. Let $$B:C$$ denote the colimit of $$A$$ . By [0011], the étale limit of the diagram induced by $$A$$ is $$C/B$$ . Then apply [001G].