You may divide Intuitionistic Propositional Logic into the negative and positive fragments. The negative fragment includes truth, conjunction, and implication while the positive fragment includes falsity and disjunction. There is an obvious duality between (truth, conjunction) and (falsity, disjunction) but implication lacks a dual. Is there a dual to implication?

Taken as a category we can make the (truth, conjunction)/(falsity, disjunction) duality explicit as (final, product)/(initial, coproduct) but I've never seen a dual to the exponential object. Is there a meaningful way to dualize the exponential object?

As a follow up there may be no way to dualize implication/exponential. If that is the case, why is implication thought to be contained in the negative fragment of IPL?