## Motivation Let $X\in S$ be a space (an object of the category of spaces), let $Sh(X)$ be the category of sheaves on the frame of opens on $X$, let $(S/X)^{et}$ denote the wide subcategory of $S/X$ with only étale morphisms. Then there is an adjoint equivalence $$(L\dashv \Gamma):(S/X)^{et}\stackrel{\Gamma}{\to}Sh(X)$$ where * $\Gamma$ sends an étale morphism $f:U\to X$ to the sheaf of local sections of $f$. * $L$ sends a sheaf on $X$ to its espace étale.