[HoTT] Recovering an equivalence from univalence in cubical type theory

[Michael Shulman <shulman@sandiego.edu>]

Let Equiv(A,B) denote the type of half-adjoint equivalences, so that
an e:Equiv(A,B) consists of five data: a function A -> B, a function B
-> A, two homotopies, and a coherence 2-path.  Using univalence, we
can make e into an identification ua(e) : A=B, and then back into an
equivalence coe(ua(e)) : Equiv(A,B), which is typally equal to e.

Now we might wonder whether coe(ua(e)) might be in fact *judgmentally*
equal to e; or at least whether this might be true of some, if not
all, of its five components.  In Book HoTT this is clearly not the
case, since univalence is posited as an axiom about which we know
nothing else.  But what about cubical type theories?  Can any of the
components of an equivalence e be recovered, up to judgmental
equality, from coe(ua(e))?  (My guess would be that at least the
function A -> B, and probably also the function B -> A, can be
recovered, but perhaps not more.)

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