proposition: Lemma adj_equiv_of_cats_is_weq_of_objects (A B : category) (HA : is_univalent A) (HB : is_univalent B) (F : [A, B]) (HF : adj_equivalence_of_cats F) : isweq (pr1 (pr1 F)).
proof: Proof. set (G := right_adjoint HF). set (et := unit_z_iso_from_adj_equivalence_of_cats HF). set (ep := counit_z_iso_from_adj_equivalence_of_cats HF). set (AAcat := is_univalent_functor_category A _ HA). set (BBcat := is_univalent_functor_category B _ HB). set (Et := isotoid _ AAcat et). set (Ep := isotoid _ BBcat ep). apply (isweq_iso _ (λ b, pr1 (right_adjoint HF) b)); intro a. apply (!toforallpaths _ _ _ (base_paths _ _ (base_paths _ _ Et)) a). now apply (toforallpaths _ _ _ (base_paths _ _ (base_paths _ _ Ep))). Defined.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Adjunctions.Core