Lemma reindexing_forward_mor_eq {a' b':C'} {x' :ob_disp reindexing_forward_ob_mor a'} {y' :ob_disp reindexing_forward_ob_mor b'} {f': C' ⟦ a', b' ⟧} {Df' : x' -->[ f'] y'} {Dg' : x' -->[ f'] y'} (e:↙ Df' = ↙ Dg') :transportf _ e ←Df' = ←Dg' -> Df'= Dg'. Proof. intro H. apply (total2_paths_f e). etrans. apply (transportf_dirprod (↙x' --> ↙ y') (λ f, double_transport ¤ (x') ¤ (y') (# F f) = f') (mor_disp ←x' ←y') Df' Dg' e). apply dirprod_paths. - apply uip. apply homset_property. - apply H. 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.