[000R] 1·e (Geometric morphisms between the little and big étale topoi).

The immersion $i:\ET/V\hookrightarrow M/V$ is left exact, continuous and cocontinuous (Artin et al., 1972). It extends into an immersion

$$ \Sum{i} : \Sh{\ET/V}\hookrightarrow \Sh{M/V}=\Sh{M}/\Yo{V} $$

which will allow us to identify the étalés over $V$ with objects of $\Sh{M}$ over $V$. $\Sum{i}$ is left exact, and has a right adjoint $i^\sharp$ (the associated “étalé” functor) which itself has a right adjoint $\Prod{i}$. We therefore have two geometric morphisms

$$ \prn{i^\sharp,\Sum{i}} : \Sh{M}/\Yo{V}\to \Sh{\ET/V}, \quad \prn{\Prod{i},i^\sharp} : \Sh{\ET/V}\to \Sh{M}/\Yo{V} $$

and the composite $i^\sharp\circ\Sum{i}$ is isomorphic to the identity.