proposition: Proposition psfunctor_psfunctor_bicat_to_fib_id_laws (F : psfunctor (cat_to_bicat C^op) bicat_of_univ_cats) : disp_nat_trans_axioms (psfunctor_psfunctor_bicat_to_fib_id_data F).
proof: Proof. intros x y f xx yy ff ; cbn -[psfunctor_id psfunctor_comp]. pose (p := nat_trans_eq_pointwise (psfunctor_linvunitor F f) yy). cbn -[psfunctor_id psfunctor_comp] in p. rewrite id_left in p. rewrite assoc'. etrans. { apply maponpaths. exact (!p). } refine (!_). etrans. { apply maponpaths. apply maponpaths_2. etrans. { do 2 apply maponpaths. etrans. { apply maponpaths. apply id_right. } apply id_right. } exact (!(nat_trans_ax (pr1 (psfunctor_id F x)) _ _ ff)). } cbn -[psfunctor_id psfunctor_comp]. pose (q := nat_trans_eq_pointwise (psfunctor_rinvunitor F f) yy). cbn -[psfunctor_id psfunctor_comp] in q. rewrite assoc'. etrans. { do 2 apply maponpaths. refine (_ @ !q). apply maponpaths_2. exact (!(id_left _)). } etrans. { apply (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat F)). } rewrite !assoc'. apply maponpaths. etrans. { apply maponpaths_2. apply psfunctor_idtoiso. } refine (_ @ !(psfunctor_idtoiso F _ _)). refine (!(pr1_idtoiso_concat _ _) @ _). do 2 apply maponpaths. refine (!(maponpathscomp0 (λ h, pr1 (#F h) yy) _ _) @ _). apply maponpaths. 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.PseudoFunctorsIntoCat UniMath.Bicategories.Transformations.PseudoTransformation UniMath.Bicategories.Transformations.Examples.PseudoTransformationIntoCat UniMath.Bicategories.Modifications.Modification UniMath.Bicategories.Modifications.Examples.ModificationIntoCat