proposition: Lemma is_functor_from_cleaving_data : is_functor fiber_functor_from_cleaving_data.
proof: Proof. split. - intro d; cbn. apply pathsinv0. apply path_to_ctr. etrans; [apply id_left_disp |]. apply pathsinv0. etrans. { apply maponpaths. apply id_right_disp. } etrans; [ apply transport_f_f |]. unfold transportb. apply maponpaths_2. apply homset_property. - intros d'' d' d ff' ff; cbn. apply pathsinv0. apply path_to_ctr. etrans; [apply mor_disp_transportf_postwhisker |]. apply pathsinv0. etrans. { apply maponpaths; apply mor_disp_transportf_prewhisker. } etrans; [apply transport_f_f |]. apply transportf_comp_lemma. apply pathsinv0. etrans; [apply assoc_disp_var |]. apply pathsinv0. apply transportf_comp_lemma. apply pathsinv0. etrans ; [ apply maponpaths, cartesian_factorisation_commutes |]. etrans ; [ apply mor_disp_transportf_prewhisker |]. apply pathsinv0. apply transportf_comp_lemma. apply pathsinv0. etrans; [ apply assoc_disp |]. apply pathsinv0. apply transportf_comp_lemma. apply pathsinv0. etrans; [ apply maponpaths_2, cartesian_factorisation_commutes |]. etrans; [ apply mor_disp_transportf_postwhisker |]. etrans. { apply maponpaths. apply assoc_disp_var. } etrans. { apply transport_f_f. } apply maponpaths_2, homset_property. Qed.
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