Would someone in !math like to help me understand Def 1.8, unity of opposite moments? https://ncatlab.org/nlab/show/Science+of+Logic

My comprehension is so poor that it’s hard for me even to frame a good question here. Is H □ ‘obviously’ isomorphic to the adjoint triple formed by the inclusion and the projection? Or is this isomorphism a further condition to the adjointness of □ and ◯? I see the symmetry, but only superficially.