Lemma restricted_adjunction_equivalence : equivalence_of_cats (full_subcat_nat_trans_is_z_iso η) (full_subcat_nat_trans_is_z_iso ε). Proof. exists (adjunction_data_from_is_left_adjoint are_adjoints_restricted_adjunction). split. - intro a. pose (isomor := make_z_iso' _ (pr2 a) : z_iso (pr1 a) (pr1 (right_adjoint are_adjoints_restricted_adjunction (left_adjoint are_adjoints_restricted_adjunction a)))). apply (iso_in_precat_is_iso_in_subcat C _ _ _ isomor). - intro b. cbn. pose (isomor := make_z_iso' _ (pr2 b) : z_iso (pr1 (left_adjoint are_adjoints_restricted_adjunction (right_adjoint are_adjoints_restricted_adjunction b))) (pr1 b)). apply (iso_in_precat_is_iso_in_subcat _ _ _ _ isomor). Defined.
Require Import UniMath.Foundations.Preamble. Require Import UniMath.Foundations.Propositions. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Subcategory.Core. Require Import UniMath.CategoryTheory.Subcategory.Full. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.whiskering.