A site is called a local site if
This appears as (Johnstone example C.3.6.3 (d)).
Since has a terminal object, the global section functor is given by evaluation on that object, hence is precomposition of sheaves with the inclusion . At the level of presheaves this has a right Kan extension functor, given by sending a set to the presheaf
This is indeed a sheaf if is covered only by the trivial cover.
local site / ∞-local site
The definition appears as example C.3.6.3 (d) in
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