Definition lan_after_pre_comp_iso_is_counit (P : [C'', D]) : z_iso_mor (lan_after_pre_comp_iso P) = counit_from_right_adjoint (is_right_adjoint_precomposition HD F') P. Proof. apply invmap_eq. apply (maponpaths (pre_whisker_in_funcat C C' D F)). apply nat_trans_eq_alt. intro c. apply colim_mor_eq. intro v. refine (colimArrowCommutes _ _ _ _ @ !_). refine (colim_mor_commute _ _ _ _ _ @ !_). apply (maponpaths #(P : C'' ⟶ D)). apply (homotweqinvweq (weq_from_fully_faithful _ _ _)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.LeftKanExtension. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.AlgebraicTheories.FundamentalTheorem.SurjectivePrecomposition. Section AdjointEquivalenceFromComp. End AdjointEquivalenceFromComp.