Definition disp_cat_eq_step_4 : (∑ (p : pr11 D₁ ╝ pr11 D₂), transportf (disp_cat_id_comp C) (total2_paths_f (pr1 p) (pr2 p)) (pr21 D₁) = pr21 D₂) ≃ ∑ (pob : ∏ (x : C), D₁ x = D₂ x), ∑ (pmor : ∏ (x y : C) (xx : D₁ x) (yy : D₁ y) (f : x --> y), xx -->[ f ] yy = eqweqmap (pob x) xx -->[ f ] eqweqmap (pob y) yy), disp_cat_id_comp_eq pob pmor. Proof. refine (invweq (totalAssociativity _) ∘ _)%weq. use weqbandf. - exact disp_cat_eq_step_4_help. - intro p. unfold disp_cat_id_comp_eq ; cbn. induction D₁ as [ [ [ D₁o D₁m ] [ D₁i D₁c ] ] D₁a ]. induction D₂ as [ [ [ D₂o D₂m ] [ D₂i D₂c ] ] D₂a ]. cbn in p. induction p as [ p₁ p₂ ]. cbn in p₁. induction p₁. cbn in p₂. induction p₂. cbn. refine (_ ∘ pathsdirprodweq)%weq. use weqdirprodf. + refine (weqonsecfibers _ _ (λ x, _) ∘ weqtoforallpaths _ _ _)%weq. exact (weqtoforallpaths _ _ _). + refine (weqonsecfibers _ _ (λ x, _) ∘ weqtoforallpaths _ _ _)%weq. refine (weqonsecfibers _ _ (λ y, _) ∘ weqtoforallpaths _ _ _)%weq. refine (weqonsecfibers _ _ (λ z, _) ∘ weqtoforallpaths _ _ _)%weq. refine (weqonsecfibers _ _ (λ f, _) ∘ weqtoforallpaths _ _ _)%weq. refine (weqonsecfibers _ _ (λ g, _) ∘ weqtoforallpaths _ _ _)%weq. refine (weqonsecfibers _ _ (λ xx, _) ∘ weqtoforallpaths _ _ _)%weq. refine (weqonsecfibers _ _ (λ yy, _) ∘ weqtoforallpaths _ _ _)%weq. refine (weqonsecfibers _ _ (λ zz, _) ∘ weqtoforallpaths _ _ _)%weq. refine (weqonsecfibers _ _ (λ ff, _) ∘ weqtoforallpaths _ _ _)%weq. exact (weqtoforallpaths _ _ _). Defined.
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.DisplayedFunctorEq. Require Import UniMath.CategoryTheory.DisplayedCats.EquivalenceOverId.