proposition: Lemma binprod_runitor {a₁ a₂ b₁ b₂ : B} (f : a₁ --> a₂) (g : b₁ --> b₂) : runitor (f ⊗₁ g) = (f ⊗₁ g ◃ (pair_1cell_id_id_invertible _ _)^-1) • pair_1cell_comp f (id₁ _) g (id₁ _) • runitor f ⊗₂ runitor g.
proof: Proof. use binprod_ump_2cell_unique_alt. - apply (pr2 B). - rewrite <- !rwhisker_vcomp. rewrite !vassocl. refine (!_). etrans. { apply maponpaths. etrans. { apply maponpaths. apply pair_2cell_pr1. } apply maponpaths_2. apply binprod_ump_2cell_pr1. } rewrite !vassocl. etrans. { do 6 apply maponpaths. rewrite !vassocr. rewrite vcomp_linv. rewrite id2_left. apply idpath. } rewrite !vassocr. rewrite <- rwhisker_lwhisker_rassociator. rewrite !vassocl. etrans. { apply maponpaths. apply maponpaths_2. apply maponpaths. apply binprod_ump_2cell_pr1. } cbn. etrans. { apply maponpaths. rewrite !vassocr. rewrite lwhisker_vcomp. rewrite !vassocl. rewrite vcomp_linv. rewrite id2_right. rewrite <- lwhisker_vcomp. apply idpath. } rewrite !vassocr. rewrite lwhisker_hcomp. rewrite triangle_l. rewrite <- rwhisker_hcomp. rewrite !vassocl. refine (_ @ id2_right _). apply maponpaths. etrans. { do 2 apply maponpaths. etrans. { apply maponpaths. rewrite !vassocr. rewrite runitor_triangle. rewrite <- vcomp_runitor. apply idpath. } rewrite !vassocr. rewrite rwhisker_vcomp. rewrite vcomp_rinv. rewrite id2_rwhisker. apply id2_left. } rewrite <- runitor_triangle. etrans. { apply maponpaths. rewrite !vassocr. rewrite lassociator_rassociator. apply id2_left. } rewrite lwhisker_vcomp. rewrite rinvunitor_runitor. apply lwhisker_id2. - rewrite <- !rwhisker_vcomp. rewrite !vassocl. refine (!_). etrans. { apply maponpaths. etrans. { apply maponpaths. apply pair_2cell_pr2. } apply maponpaths_2. apply binprod_ump_2cell_pr2. } rewrite !vassocl. etrans. { do 6 apply maponpaths. rewrite !vassocr. rewrite vcomp_linv. rewrite id2_left. apply idpath. } rewrite !vassocr. rewrite <- rwhisker_lwhisker_rassociator. rewrite !vassocl. etrans. { apply maponpaths. apply maponpaths_2. apply maponpaths. apply binprod_ump_2cell_pr2. } cbn. etrans. { apply maponpaths. rewrite !vassocr. rewrite lwhisker_vcomp. rewrite !vassocl. rewrite vcomp_linv. rewrite id2_right. rewrite <- lwhisker_vcomp. apply idpath. } rewrite !vassocr. rewrite lwhisker_hcomp. rewrite triangle_l. rewrite <- rwhisker_hcomp. rewrite !vassocl. refine (_ @ id2_right _). apply maponpaths. etrans. { do 2 apply maponpaths. etrans. { apply maponpaths. rewrite !vassocr. rewrite runitor_triangle. rewrite <- vcomp_runitor. apply idpath. } rewrite !vassocr. rewrite rwhisker_vcomp. rewrite vcomp_rinv. rewrite id2_rwhisker. apply id2_left. } rewrite <- runitor_triangle. etrans. { apply maponpaths. rewrite !vassocr. rewrite lassociator_rassociator. apply id2_left. } rewrite lwhisker_vcomp. rewrite rinvunitor_runitor. apply lwhisker_id2. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Categories.StandardCategories UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Equivalences.FullyFaithful UniMath.CategoryTheory.PrecategoryBinProduct UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Core.Unitors UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Core.AdjointUnique UniMath.Bicategories.Core.Univalence