Proposition is_equiv_over_id_to_ff : disp_functor_ff LL. Proof. intros x y xx yy f. use isweq_iso. - cbn ; intro ff. refine (transportf (λ z, _ -->[ z ] _) _ (η x xx ;; ♯ RR ff ;; inv_mor_disp_from_z_iso (is_z_iso_unit_over_id HLL y yy))). abstract (cbn ; rewrite id_left, id_right ; apply idpath). - abstract (intro ff ; cbn ; etrans ; [ apply maponpaths ; apply maponpaths_2 ; exact (disp_nat_trans_ax_var η ff) | ] ; cbn ; rewrite mor_disp_transportf_postwhisker ; rewrite transport_f_f ; rewrite assoc_disp_var ; rewrite transport_f_f ; etrans ; [ do 2 apply maponpaths ; exact (inv_mor_after_z_iso_disp (is_z_iso_unit_over_id (pr2 HLL) y yy)) | ] ; cbn ; unfold transportb ; rewrite mor_disp_transportf_prewhisker ; rewrite transport_f_f ; rewrite id_right_disp ; unfold transportb ; rewrite transport_f_f ; apply transportf_set ; apply homset_property). - abstract (cbn ; intro ff ; rewrite (disp_functor_transportf _ LL) ; rewrite disp_functor_comp ; unfold transportb ; rewrite transport_f_f ; rewrite disp_functor_comp ; unfold transportb ; rewrite mor_disp_transportf_postwhisker ; rewrite transport_f_f ; etrans ; [ do 2 apply maponpaths ; apply triangle_1_over_id_alt | ] ; rewrite assoc_disp_var ; rewrite transport_f_f ; etrans ; [ do 2 apply maponpaths ; apply (disp_nat_trans_ax (counit_over_id HLL)) | ] ; unfold transportb ; rewrite mor_disp_transportf_prewhisker ; rewrite transport_f_f ; cbn ; rewrite assoc_disp ; unfold transportb ; rewrite transport_f_f ; etrans ; [ apply maponpaths ; apply maponpaths_2 ; exact (triangle_1_over_id HLL x xx) | ] ; unfold transportb ; rewrite mor_disp_transportf_postwhisker ; rewrite transport_f_f ; rewrite id_left_disp ; unfold transportb ; rewrite transport_f_f ; apply transportf_set ; apply homset_property) ; rewrite triangle_1_over_id_alt. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq.