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. 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.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Adjunctions.Core.