Could not include topos theory - contents
A site is called a local site if
it has a terminal object $*$;
the only covering family of $*$ is the trivial cover.
This appears as (Johnstone example C.3.6.3 (d)).
The category of sheaves $Sh(C)$ on a local site $C$ is a local topos.
Since $C$ has a terminal object, the global section functor $Sh(C) \to Set$ is given by evaluation on that object, hence is precomposition of sheaves with the inclusion $* \to C$. At the level of presheaves this has a right Kan extension functor, given by sending a set $S$ to the presheaf
This is indeed a sheaf if $*$ is covered only by the trivial cover.
See (Johnstone example C.3.6.3 (d)).
and
local site / ∞-local site
The definition appears as example C.3.6.3 (d) in
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