proposition: Proposition psfunctor_fib_to_psfunctor_unit_equiv_unit_laws (D : disp_univalent_category C) (HD : cleaving D) : disp_nat_trans_axioms (psfunctor_fib_to_psfunctor_unit_equiv_unit_data D HD).
proof: Proof. intros x y f xx yy ff. cbn. unfold psfunctor_fib_to_psfunctor_unit_equiv_unit_data. rewrite !mor_disp_transportf_postwhisker. unfold transportb. rewrite !transport_f_f. use (cartesian_factorisation_unique (HD _ _ _ _)). rewrite !mor_disp_transportf_postwhisker. etrans. { rewrite assoc_disp_var. 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. etrans. { do 2 apply maponpaths. rewrite assoc_disp. apply idpath. } unfold transportb. rewrite !mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite !mor_disp_transportf_postwhisker. rewrite !mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite assoc_disp_var. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite !mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite id_right_disp. unfold transportb. rewrite !mor_disp_transportf_prewhisker. rewrite transport_f_f. apply idpath. } refine (!_). etrans. { rewrite (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat (indexed_cat_to_psfunctor (cleaving_to_indexed_cat D HD)))). cbn. etrans. { apply maponpaths_2. do 2 apply maponpaths. apply idtoiso_fiber_category. } rewrite !mor_disp_transportf_postwhisker. rewrite transport_f_f. rewrite assoc_disp_var. rewrite transport_f_f. rewrite assoc_disp_var. rewrite transport_f_f. etrans. { do 3 apply maponpaths. apply idtoiso_disp_cartesian_lift. } rewrite !mor_disp_transportf_prewhisker. 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. etrans. { apply maponpaths. apply maponpaths_2. rewrite assoc_disp_var. rewrite cartesian_factorisation_commutes. apply idpath. } rewrite mor_disp_transportf_prewhisker. rewrite !mor_disp_transportf_postwhisker. rewrite !transport_f_f. rewrite assoc_disp_var. rewrite transport_f_f. rewrite assoc_disp_var. rewrite mor_disp_transportf_prewhisker. 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 idpath. } apply maponpaths_2. apply homset_property. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.OppositeCategory.Core UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.IndexedCategories.IndexedCategory UniMath.CategoryTheory.IndexedCategories.IndexedFunctor UniMath.CategoryTheory.IndexedCategories.IndexedTransformation UniMath.CategoryTheory.IndexedCategories.FibrationToIndexedCategory UniMath.CategoryTheory.IndexedCategories.CartesianToIndexedFunctor UniMath.CategoryTheory.IndexedCategories.NatTransToIndexed UniMath.CategoryTheory.IndexedCategories.IndexedCategoryToFibration UniMath.CategoryTheory.IndexedCategories.IndexedFunctorToCartesian UniMath.CategoryTheory.IndexedCategories.IndexedTransformationToTransformation UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Discreteness UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.Univalence UniMath.Bicategories.Core.EquivToAdjequiv UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Examples.DiscreteBicat UniMath.Bicategories.Core.Examples.FibSlice UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat UniMath.Bicategories.PseudoFunctors.PseudoFunctor UniMath.Bicategories.PseudoFunctors.Examples.Identity UniMath.Bicategories.PseudoFunctors.Examples.Composition UniMath.Bicategories.PseudoFunctors.Examples.PseudoFunctorsIntoCat UniMath.Bicategories.Transformations.PseudoTransformation UniMath.Bicategories.Transformations.Examples.PseudoTransformationIntoCat UniMath.Bicategories.Modifications.Modification UniMath.Bicategories.Modifications.Examples.ModificationIntoCat UniMath.Bicategories.Grothendieck.FibrationToPseudoFunctor UniMath.Bicategories.Grothendieck.PseudoFunctorToFibration