Morphisms and comorphisms of sites I -- Double categories of sites

We arrange morphisms and comorphisms of sites as the horizontal and vertical cells of a double category of sites; using the formalism of extensions and restrictions of presheaves, we explains how one can define a sheafification double functor from this double category to the quintet double category of Grothendieck topoi. We describe properties of this double functor and recover some classical results of topos theory through a new notion of locally exact square, generalizing exact squares in the presence of topologies. We also describe a 2-comonad on $\Cat$ for which lax morphisms of coalgebras are morphisms of sites and colax morphisms are comorphisms of sites, explaining the arrangement as a double category.