By the adjoint functor theorem, one can demonstrate that every frame is a complete Heyting algebra, and vice versa, so far as the underlying poset goes.

However, morphisms of frames needn’t preserve exponential objects or infinitary meets, as would most naturally be required of morphisms of complete Heyting algebras. Also, when considering large lattices which are only small-complete, then frames and complete Heyting algebras are different objects.