proposition: Proposition psfunctor_psfunctor_bicat_to_fib_comp_axioms {F G H : psfunctor (cat_to_bicat C^op) bicat_of_univ_cats} (n₁ : pstrans F G) (n₂ : pstrans G H) : disp_nat_trans_axioms (psfunctor_psfunctor_bicat_to_fib_comp_data n₁ n₂).
proof: Proof. intros x y f xx yy ff ; cbn -[psfunctor_id psfunctor_comp]. refine (!_). etrans. { apply maponpaths. apply maponpaths_2. do 3 apply maponpaths. refine (id_left _ @ _). apply maponpaths. refine (id_left _ @ _). apply id_right. } etrans. { apply (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat H)). } etrans. { do 2 apply maponpaths_2. refine (!_). apply (nat_trans_ax (pr1 (psfunctor_id H x))). } refine (!_). etrans. { do 3 apply maponpaths_2. apply functor_comp. } do 3 refine (assoc' _ _ _ @ _). refine (!_). do 3 refine (assoc' _ _ _ @ _). apply maponpaths. refine (assoc' _ _ _ @ _). do 2 apply maponpaths. etrans. { refine (assoc _ _ _ @ _). apply maponpaths_2. refine (_ @ !(nat_trans_eq_pointwise (psfunctor_rinvunitor H f) _)). refine (_ @ assoc _ _ _). exact (!(id_left _)). } etrans. { apply maponpaths_2. apply psfunctor_idtoiso. } refine (!(pr1_idtoiso_concat _ _) @ _). refine (!_). etrans. { refine (_ @ !(nat_trans_eq_pointwise (psfunctor_linvunitor H f) _)). refine (_ @ assoc _ _ _). exact (!(id_left _)). } etrans. { apply psfunctor_idtoiso. } do 2 apply maponpaths. refine (_ @ maponpathscomp0 (λ z, pr1 (#H z) _) _ _). 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