Pairs of adjoint functors between the categories of sheaves appear in varius setups, e.g. geometric morphisms of topoi, abelian categories of quasicoherent sheaves on schemes, bounded derived categories of coherent sheaves on varieties.
For example, the right adjoint part of any geometric morphism
of toposes is called a direct image.
Specifically for Grothendieck toposes: a morphism of sites induces a geometric morphism of Grothendieck toposes
between the categories of sheaves on the sites, with
the direct image
and its left adjoint: the inverse image.
Given a morphism of sites coming from a functor , the direct image operation on presheaves is the functor
The restriction of this operation to sheaves, which respects sheaves, is the direct image of sheaves
For a site with a terminal object, let the morphism of sites be the canonical morphism .
The direct image is the global sections functor;
the inverse image is the constant sheaf functor;
the left adjoint to is , the functor of geometric connected components (see homotopy group of an ∞-stack).
See
for the moment.