Proposition univcat_profunctor_double_bicat_id_comp_square_laws : double_bicat_id_comp_square_laws univcat_profunctor_ver_bicat_sq_id_comp_whisker. Proof. repeat split. - intros C₁ C₂ C₃ C₄ C₅ C₆ C₇ C₈ C₀ P₁ P₂ Q₁ Q₂ R₁ R₂ F₁ F₂ G₁ G₂ H₁ H₂ τ₁ τ₂ θ₁ θ₂. use eq_profunctor_square ; intros z x h. use mor_from_comp_profunctor_ob_eq. clear h ; cbn in *. intros y h h'. etrans. { apply maponpaths. exact (comp_v_profunctor_square_mor_comm τ₁ τ₂ z x y h h'). } etrans. { exact (comp_v_profunctor_square_mor_comm θ₁ θ₂ (H₁ z) (F₁ x) (G₁ y) (τ₁ y x h) (τ₂ z y h')). } refine (!_). etrans. { exact (comp_v_profunctor_square_mor_comm (comp_h_profunctor_square τ₁ θ₁) (comp_h_profunctor_square τ₂ θ₂) z x y h h'). } cbn. apply idpath. - intro C. use eq_profunctor_square ; intros y x h ; cbn. apply idpath. - intros C₁ C₂ C₃ P Q. use eq_profunctor_square ; intros z x h. use mor_from_comp_profunctor_ob_eq. clear h. intros y h h' ; cbn in *. etrans. { exact (comp_v_profunctor_square_mor_comm (id_h_profunctor_square P) (id_h_profunctor_square Q) z x y h h'). } cbn. apply idpath. - intros C₁ C₂ C₃ P Q. use eq_profunctor_square ; intros z x h ; cbn in *. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.Categories.HSET.SetCoends. Require Import UniMath.CategoryTheory.Profunctors.Core. Require Import UniMath.CategoryTheory.Profunctors.Examples. Require Import UniMath.CategoryTheory.Profunctors.Squares. Require Import UniMath.CategoryTheory.Profunctors.Transformation. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.ProfunctorTwosidedDispCat. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.UnivalenceOp. Require Import UniMath.Bicategories.Core.Examples.OpCellBicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.Core.