proposition: Proposition fiber_functor_from_cleaving_identity_laws : is_nat_trans _ _ fiber_functor_from_cleaving_identity_data.
proof: Proof. intros xx yy f ; cbn. unfold fiber_functor_from_cleaving_identity_data. use (cartesian_factorisation_unique (HD x x (identity x) yy)). rewrite !mor_disp_transportf_postwhisker. rewrite assoc_disp_var. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. unfold transportb. rewrite mor_disp_transportf_prewhisker. rewrite !transport_f_f. rewrite id_right_disp. unfold transportb. rewrite transport_f_f. rewrite assoc_disp_var. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite mor_disp_transportf_prewhisker. rewrite !transport_f_f. rewrite assoc_disp. unfold transportb. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite mor_disp_transportf_postwhisker. rewrite transport_f_f. rewrite id_left_disp. unfold transportb. rewrite transport_f_f. apply maponpaths_2. apply homset_property. Defined.
imports: UniMath.Foundations.Sets UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.MonoEpiIso UniMath.CategoryTheory.Categories.HSET.Univalence UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Presheaf UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Examples.Opposite