Definition adjoint_equivalence_1_from_comp : adj_equivalence_of_cats (pre_comp_functor (C := D) F'). Proof. use adj_equivalence_from_right_adjoint. - apply (is_right_adjoint_precomposition HD). - intro P. exact (z_iso_is_z_isomorphism (pre_comp_after_lan_iso _ HF' _ HD P)). - intro P. exact (is_z_isomorphism_path (lan_after_pre_comp_iso_is_counit P) (z_iso_is_z_isomorphism (lan_after_pre_comp_iso P))). 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.