Definition representable_comp_inv2cell {X Y Z: C} (f : X --> Y) (g : Y --> Z) : invertible_modification_data (comp_pstrans (representable1 C_is_univalent_2_1 f) (representable1 C_is_univalent_2_1 g)) (representable1 C_is_univalent_2_1 (f · g)). Proof. intro W. use make_invertible_2cell. - use make_nat_trans. + intro h. cbn in *. apply rassociator. + abstract (intros V U h; cbn in *; apply rwhisker_rwhisker_alt). - use make_is_invertible_2cell. + use make_nat_trans. * intro h. cbn in *. apply lassociator. * abstract (intros h k η; cbn in *; symmetry; apply rwhisker_rwhisker). + use nat_trans_eq. { exact (pr2 C W Z). } abstract (intro h; cbn in *; apply rassociator_lassociator). + use nat_trans_eq. { exact (pr2 C W Z). } abstract (intro h; cbn in *; apply lassociator_rassociator). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.PseudoFunctors.Display.Base. Require Import UniMath.Bicategories.PseudoFunctors.Display.Map1Cells. Require Import UniMath.Bicategories.PseudoFunctors.Display.Map2Cells. Require Import UniMath.Bicategories.PseudoFunctors.Display.Identitor. Require Import UniMath.Bicategories.PseudoFunctors.Display.Compositor. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.Core.Examples.OpMorBicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.PseudoFunctors.Representable. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Modifications.Modification.