Proposition iterated_slice_pb_nat_trans_laws : is_nat_trans _ _ iterated_slice_pb_nat_trans_data. Proof. intros g₁ g₂ p. use eq_mor_cod_fib. rewrite !comp_in_cod_fib. refine (id_right _ @ _ @ !(id_left _)). cbn -[cod_pb]. etrans. { apply maponpaths. exact (maponpaths dom_mor (cod_fiber_functor_on_mor PB₁ h p)). } refine (!_). etrans. { apply maponpaths. exact (cod_fiber_functor_on_mor PB (dom_mor h) _). } use (MorphismsIntoPullbackEqual (isPullback_Pullback _)). - refine (PullbackArrow_PullbackPr1 _ _ _ _ _ @ _ @ !(PullbackArrow_PullbackPr1 _ _ _ _ _)). rewrite comp_in_cod_fib. cbn. apply idpath. - exact (PullbackArrow_PullbackPr2 _ _ _ _ _ @ !(PullbackArrow_PullbackPr2 _ _ _ _ _)). 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.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Limits.PreservationProperties. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodColimits.