Definition counit_z_iso_from_adj_equivalence_of_cats : z_iso (C:=[B, B]) (functor_composite (right_adjoint HF) F) (functor_identity B). Proof. exists (counit_from_left_adjoint HF). apply nat_trafo_z_iso_if_pointwise_z_iso. intro c. apply (pr2 (pr2 HF)). 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.