Could not include topos theory - contents
The terminal object in the site represents the terminal presheaf on , which is the presheaf constant on the point. By the discussion at locally connected site we have that every constant presheaf is a sheaf over , hence the terminal object of is also represented by the terminal object in the site, and we just write “” for all these terminal objects.
By the discussion there, the left adjoint in the sheaf topos over a locally connected site is given by the colimit functor . The colimit over a representable functor is always the point (this is the (co)-Yoneda lemma in slight disguise), hence indeed .