proposition: Proposition psfunctor_fib_to_psfunctor_counit_is_pstrans : is_pstrans psfunctor_fib_to_psfunctor_counit_data.
proof: Proof. refine (_ ,, _ ,, _). - intros F₁ F₂ n₁ n₂ m. Opaque comp_psfunctor. use modification_eq. Transparent comp_psfunctor. intro x. use nat_trans_eq ; [ apply homset_property | ]. intros xx. refine (id_right _ @ _). cbn -[psfunctor_id]. refine (_ @ !(id_left _)). refine (!_). refine (_ @ id_right _). refine (assoc' _ _ _ @ _). apply maponpaths. exact (nat_trans_eq_pointwise (vcomp_rinv (psfunctor_id F₂ x)) (pr1 (pr111 n₂ x) xx)). - intros F. Opaque comp_psfunctor. use modification_eq. Transparent comp_psfunctor. intro x. use nat_trans_eq ; [ apply homset_property | ]. intros xx. refine (id_right _ @ _). cbn. refine (!_). etrans. { apply maponpaths_2. apply id_left. } refine (id_left _ @ _). etrans. { apply maponpaths_2. apply (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat F)). } unfold psfunctor_psfunctor_bicat_to_fib_id_data. refine (_ @ nat_trans_eq_pointwise (vcomp_rinv (psfunctor_id F x)) xx). cbn -[psfunctor_id]. do 3 refine (assoc' _ _ _ @ _). apply maponpaths. refine (_ @ id_left _). do 2 refine (assoc _ _ _ @ _). apply maponpaths_2. etrans. { apply maponpaths_2. etrans. { pose (p := nat_trans_eq_pointwise (psfunctor_linvunitor F (identity x)) xx). cbn -[psfunctor_id psfunctor_comp] in p. refine (_ @ !p). apply maponpaths_2. exact (!(id_left _)). } apply psfunctor_idtoiso. } refine (!(pr1_idtoiso_concat _ _) @ _ @ idtoiso_idpath _). do 2 apply maponpaths. refine (!(maponpathscomp0 (λ h, pr1 (#(F : psfunctor _ _) h) xx) _ _) @ _). refine (_ @ @maponpaths_idpath _ _ (λ h, pr1 (#(F : psfunctor _ _) h) xx) _). apply maponpaths. apply homset_property. - intros F₁ F₂ F₃ τ θ. Opaque comp_psfunctor. use modification_eq. Transparent comp_psfunctor. intro x. use nat_trans_eq ; [ apply homset_property | ]. intros xx. refine (id_right _ @ _). cbn. refine (!_). etrans. { apply maponpaths_2. do 3 refine (id_right _ @ _). etrans. { apply maponpaths. apply functor_id. } apply id_right. } refine (id_left _ @ _). unfold psfunctor_psfunctor_bicat_to_fib_comp_data. cbn -[psfunctor_id]. etrans. { apply maponpaths_2. apply (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat F₃)). } refine (_ @ nat_trans_eq_pointwise (vcomp_rinv (psfunctor_id F₃ x)) _). cbn -[psfunctor_id]. do 3 refine (assoc' _ _ _ @ _). apply maponpaths. refine (_ @ id_left _). do 2 refine (assoc _ _ _ @ _). apply maponpaths_2. etrans. { apply maponpaths_2. etrans. { pose (p := nat_trans_eq_pointwise (psfunctor_linvunitor F₃ (identity x)) (pr1 (pr111 θ x) (pr1 (pr111 τ x) xx))). cbn -[psfunctor_id psfunctor_comp] in p. refine (_ @ !p). apply maponpaths_2. exact (!(id_left _)). } apply psfunctor_idtoiso. } refine (!(pr1_idtoiso_concat _ _) @ _ @ idtoiso_idpath _). do 2 apply maponpaths. refine (!(maponpathscomp0 (λ h, pr1 (#(F₃ : psfunctor _ _) h) _) _ _) @ _). refine (_ @ @maponpaths_idpath _ _ (λ h, pr1 (#(F₃ : psfunctor _ _) h) _) _). apply maponpaths. apply homset_property. Opaque comp_psfunctor. 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