proposition: Proposition cell_are_companions_id {B : verity_double_bicat} (H : vertically_saturated B) {x y : B} {h : x --> y} {v : x -|-> y} (c : are_companions h v) : cell_are_companions H (id2 h) c c = id2 v.
proof: Proof. unfold cell_are_companions. rewrite <- (square_to_vertical_cell_id H). apply maponpaths. rewrite dwhisker_square_id. rewrite are_companions_left'. rewrite <- rwhisker_square_comp. rewrite rinvunitor_runitor. rewrite rwhisker_square_id. rewrite <- lwhisker_square_comp. rewrite linvunitor_lunitor. rewrite lwhisker_square_id. apply idpath. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Unitors UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Core.EquivToAdjequiv UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.DoubleCategories.DoubleBicat.VerityDoubleBicat UniMath.Bicategories.DoubleCategories.DoubleBicat.DerivedLaws UniMath.Bicategories.DoubleCategories.DoubleBicat.CellsAndSquares UniMath.Bicategories.DoubleCategories.DoubleBicat.CompanionPairs