Lemma reindexing_forward_comp {a' b' c': C'} {x' : ob_disp reindexing_forward_ob_mor a'} {y' : ob_disp reindexing_forward_ob_mor b'} {z' : ob_disp reindexing_forward_ob_mor c'} {f':C' ⟦ a', b' ⟧} {g':C' ⟦ b', c' ⟧} {Df': x' -->[ f'] y'} {Dg': y' -->[ g'] z'} :f' · g' = double_transport (¤ x') (¤ z') (# F (↙ Df' · ↙ Dg')). Proof. destruct (¤ Df'), (¤ Dg'). destruct (¤ x'), (¤ y'), (¤ z'). cbn. apply pathsinv0. apply functor_comp. 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.