  We present a unified categorical treatment of completeness theorems for
several classical and intuitionistic infinitary logics with a proposed
axiomatization. This provides new completeness theorems and subsumes previous
ones by G\"odel, Kripke, Beth, Karp, Joyal, Makkai and Fourman/Grayson. As an
application we prove, using large cardinals assumptions, the disjunction and
existence properties for infinitary intuitionistic first-order logics.
