Spahn sheaf on a sheaf (Rev #1, changes)

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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

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.