Lemma precompLU_eso : disp_functor_disp_ess_split_surj precompLU. Proof. intros G GG. use tpair. - intro d. use factor_through_squash. { exact (∑ a : C, z_iso (H a) d). } { apply homset_property. } 2: exact (H_eso d). intro cd. induction (isotoid _ Duniv (pr2 cd)). refine (GG (pr1 cd) @ _). simpl. rewrite (functor_id (pr1 G)). rewrite id_right. do 3 rewrite assoc'. do 2 apply maponpaths. rewrite <- (functor_comp (pr1 G)). apply maponpaths. simpl. refine (_ @ ! TransportedLeftUnitorOnOb (pr1 cd)). apply maponpaths_2. unfold LiftPreservesPretensor. simpl. rewrite ! id_left. rewrite ! id_right. rewrite (functor_id H). rewrite id_right. etrans. 2: { apply maponpaths_2. apply (! functor_id _ _). } apply (! id_left _). - exists tt. exists tt. split ; apply isapropunit. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.PrecompEquivalence. Require Import UniMath.CategoryTheory.catiso. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.TotalCategoryFacts. Require Import UniMath.CategoryTheory.Monoidal.RezkCompletion.LiftedTensor. Require Import UniMath.CategoryTheory.Monoidal.RezkCompletion.LiftedTensorUnit.