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Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.

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a “self-dual” adjunction

Is there a name for $(U,\eta)$ such that $(\eta, \eta^{op}):U^{op}\dashv U$ (is an adjunction). To clarify — $C:category$, $(I,I^{op})$ is the contravariant isomorphism with $I:C^{op}\to C$, $U:C^{op} …