proposition: Proposition cod_fiber_functor_from_cleaving_comp_inv {x y z : C} (f : x --> y) (g : y --> z) (φ : C/z) : fiber_functor_from_cleaving_comp_inv HD g f φ = cod_fiber_functor_from_cleaving_comp_mor_inv f g φ.
proof: Proof. use eq_mor_cod_fib. use (MorphismsIntoPullbackEqual (isPullback_Pullback _)). - cbn. unfold cartesian_cod_disp_map ; cbn. refine (PullbackArrow_PullbackPr1 _ _ _ _ _ @ _). rewrite PullbackArrow_PullbackPr1. use (MorphismsIntoPullbackEqual (isPullback_Pullback _)). + refine (PullbackArrow_PullbackPr1 _ _ _ _ _ @ _). rewrite PullbackArrow_PullbackPr1. rewrite pr1_transportf, transportf_const ; cbn. apply idpath. + refine (PullbackArrow_PullbackPr2 _ _ _ _ _ @ _). rewrite PullbackArrow_PullbackPr2. rewrite id_left. apply idpath. - cbn. unfold cartesian_cod_disp_map ; cbn. refine (PullbackArrow_PullbackPr2 _ _ _ _ _ @ _). rewrite PullbackArrow_PullbackPr2. apply id_right. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Codomain