Question

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 $I:C^{op}\to C$, $U:C^{op}\to C,\ U^{op}:=I^{op}\circ U\circ I^{op},\ U^{op}:C\to C^{op}$, $\eta:id(C)\to U\circ U^{op},\ \eta^{op}:= I^{op}\eta I,\ \eta^{op}:U^{op}\circ U\to id(C^{op})$. E.g. in CCC, contravariant exponential functor and $\eta(a):=\lambda(x:a) f.f\ x$ is such an adjunction.

Answer 1

(I'm a bit confused by your notation (what is $I$?), but if you mean what I think you mean...)

I don't think there is an 'official' name for these things, but I've seen the term 'self-adjoint' used, sometimes qualified by 'on the left' or 'on the right' according to whether $U \dashv U^{op}$ or $U^{op} \dashv U$. See e.g. Mac Lane & Moerdijk, Sheaves in Geometry and Logic, chapter IV, section 5.

I believe it was Manes who observed that the power-object functor $P \colon E^{op} \to E$ of an elementary topos $E$ is not only self-adjoint on the right but monadic, with the corollary that toposes have finite colimits. (See loc. cit.)

Hayo Thielecke has studied self-adjunctions as a way to understand the notion of 'continuations' in computer science. See his Edinburgh Ph.D. thesis.