proposition: Lemma presheaf_to_yoneda_ob_pstrans_is_pstrans : is_pstrans presheaf_to_yoneda_ob_pstrans_data.
proof: Proof. repeat split. - intros Y₁ Y₂ g₁ g₂ α. apply nat_trans_eq. { apply homset_property. } intro h ; cbn in *. unfold presheaf_to_yoneda_ob_pstrans_functor_ob, presheaf_to_yoneda_ob_pstrans_functor_mor, presheaf_to_yoneda_ob_pstrans_nat_trans_data. pose (psfunctor_lwhisker F h α). pose (nat_trans_eq_pointwise p x) as q. exact (!q). - intros Y. apply nat_trans_eq. { apply homset_property. } intro h ; cbn in *. unfold presheaf_to_yoneda_ob_pstrans_functor_ob, presheaf_to_yoneda_ob_pstrans_functor_mor, presheaf_to_yoneda_ob_pstrans_nat_trans_data. refine (!_). etrans. { etrans. { apply maponpaths_2. apply id_left. } etrans. { apply id_left. } exact (nat_trans_eq_pointwise (psfunctor_rinvunitor F h) x). } cbn -[psfunctor_id psfunctor_comp]. apply maponpaths_2. apply id_left. - intros Y₁ Y₂ Y₃ g₁ g₂. apply nat_trans_eq. { apply homset_property. } intro h ; cbn in *. unfold presheaf_to_yoneda_ob_pstrans_functor_ob, presheaf_to_yoneda_ob_pstrans_functor_mor, presheaf_to_yoneda_ob_pstrans_nat_trans_data. refine (!_). etrans. { etrans. { apply maponpaths_2. etrans. { apply id_right. } apply maponpaths_2. etrans. { apply id_right. } apply id_left. } exact (nat_trans_eq_pointwise (psfunctor_rassociator F h g₁ g₂) x). } simpl. etrans. { apply maponpaths_2. apply id_left. } apply idpath. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Core.Examples.OpMorBicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.EquivToAdjequiv UniMath.Bicategories.Core.Unitors UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Morphisms.Properties UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.Univalence UniMath.Bicategories.Core.UnivalenceOp UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat UniMath.Bicategories.PseudoFunctors.PseudoFunctor UniMath.Bicategories.PseudoFunctors.Properties UniMath.Bicategories.Transformations.PseudoTransformation UniMath.Bicategories.Modifications.Modification UniMath.Bicategories.PseudoFunctors.Yoneda UniMath.Bicategories.PseudoFunctors.Representable UniMath.Bicategories.Core.Examples.Image UniMath.Bicategories.PseudoFunctors.Examples.CorestrictImage