It is well-known the correspondence between intuitionist propositional logic (IPL) and cartesian closed categories (CCCs). From what I understand, under this correspondence, two proofs of IPL are considered equal if they denote the same morphism in the corresponding CCC.
It is also well-known that the equivalent categories to CCCs for Boolean algebras are actually posets. So it seems that any two proofs in Boolean algebra are equivalent. Why is this the case? From the logical side, in what sense are they equivalent? What's wrong with my intuition about "equivalence" of proofs?