Definition kleisli_triple_disp_cat_data : disp_cat_data bicat_of_univ_cats. Proof. use tpair. - use tpair ; cbn. + exact kleisli_triple. + exact @kleisli_triple_on_functor. - split ; cbn. + exact @kleisli_triple_on_identity_functor. + intros C₁ C₂ C₃ F₁ F₂ M₁ M₂ M₃ MF₁ MF₂. use tpair. * exact (λ X, z_iso_comp (pr1 MF₂ (F₁ X)) (functor_on_z_iso F₂ (pr1 MF₁ X))). * split. ** abstract (intros A ; cbn; rewrite (pr12 MF₁); rewrite functor_comp; rewrite (pr12 MF₂); rewrite assoc; apply idpath). ** abstract (intros A B f ; simpl; rewrite (pr22 MF₁); rewrite !functor_comp; rewrite (pr22 MF₂); rewrite !functor_comp; rewrite (inv_from_z_iso_z_iso_comp (pr1 MF₂ (F₁ A))); rewrite (inv_from_z_iso_z_iso_comp (pr1 MF₂ (F₁ B))); rewrite <- !functor_on_inv_from_z_iso; rewrite !assoc; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.