Lemma reindexing_forward_ob_eq {c' : C'} {x' :ob_disp reindexing_forward_ob_mor c'} {y' :ob_disp reindexing_forward_ob_mor c'} (e0 : ↙x' = ↙y') : transportf (λ c, F c = c') e0 ¤x' = ¤y' -> transportf _ e0 ←x' = ←y' -> x' = y'. Proof. intros e1 e2. apply (total2_paths_f e0). etrans. exact (transportf_dirprod C (λ c, F c = c') D x' y' e0). apply dirprod_paths. - exact e1. - exact e2. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Isos. 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.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.StreetFibration.