Spahn sheaf on a sheaf (Rev #2)

Motivation

Let $X\in H$ be a space (an object of a category $H$ of spaces), let $Sh(X)$ be the category of sheaves on the frame of opens on $X$, let $(H/X)^{et}$ denote the wide subcategory of $H/X$ 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.

Très petit topos

We wish to clarify in which sense also the $(\infty,1)$- topos $(H/X)^{fet}$ can be regarded as an $(\infty,1)$-sheaftopos on $X$.