proposition: Proposition comp_functor_unit_laws : is_nat_trans _ _ comp_functor_unit_data.
proof: Proof. intros g₁ g₂ h. use eq_mor_cod_fib. cbn -[fiber_category]. rewrite !comp_in_cod_fib. cbn. use (MorphismsIntoPullbackEqual (isPullback_Pullback _)). - rewrite !assoc'. rewrite PullbackArrow_PullbackPr1. refine (!_). etrans. { apply maponpaths. apply (PullbackArrow_PullbackPr1 (HC y (cod_dom g₂) x (cod_mor g₂ · f) f)). } rewrite transportf_cod_disp. cbn. rewrite !assoc. rewrite PullbackArrow_PullbackPr1. rewrite id_left, id_right. apply idpath. - rewrite !assoc'. rewrite PullbackArrow_PullbackPr2. refine (!_). etrans. { apply maponpaths. apply (PullbackArrow_PullbackPr2 (HC y (cod_dom g₂) x (cod_mor g₂ · f) f)). } cbn. rewrite !assoc. rewrite PullbackArrow_PullbackPr2. refine (_ @ !(mor_eq h)). apply id_right. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Codomain UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod UniMath.CategoryTheory.DisplayedCats.DependentSums