proposition: Proposition univalent_profunctor_double_bicat_id_comp_square_laws : double_bicat_id_comp_square_laws univalent_profunctor_ver_bicat_sq_id_comp_whisker.
proof: 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.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Setcategories UniMath.CategoryTheory.Categories.CategoryOfSetCategories UniMath.CategoryTheory.Categories.HSET.All UniMath.CategoryTheory.Categories.HSET.SetCoends UniMath.CategoryTheory.Profunctors.Core UniMath.CategoryTheory.Profunctors.Examples UniMath.CategoryTheory.Profunctors.Squares UniMath.CategoryTheory.Profunctors.Transformation UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.ProfunctorTwosidedDispCat UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.Univalence UniMath.Bicategories.Core.UnivalenceOp UniMath.Bicategories.Core.Examples.OpCellBicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.DoubleCategories.DoubleBicat.Core