proposition: Lemma binprod_lwhisker {a₁ a₂ a₃ b₁ b₂ b₃ : B} (f₁ : a₁ --> a₂) (f₂ : b₁ --> b₂) {g₁ h₁ : a₂ --> a₃} {g₂ h₂ : b₂ --> b₃} (τ₁ : g₁ ==> h₁) (τ₂ : g₂ ==> h₂) : pair_1cell_comp f₁ g₁ f₂ g₂ • ((f₁ ◃ τ₁) ⊗₂ (f₂ ◃ τ₂)) = ((f₁ ⊗₁ f₂) ◃ (τ₁ ⊗₂ τ₂)) • pair_1cell_comp f₁ h₁ f₂ h₂.
proof: Proof. use binprod_ump_2cell_unique_alt. - apply (pr2 B). - rewrite <- !rwhisker_vcomp. etrans. { apply maponpaths. apply pair_2cell_pr1. } etrans. { apply maponpaths_2. apply binprod_ump_2cell_pr1. } rewrite !vassocl. use vcomp_move_R_pM ; [ is_iso | ] ; cbn. refine (!_). etrans. { rewrite !vassocr. rewrite <- rwhisker_lwhisker. rewrite !vassocl. apply maponpaths_2. etrans. { apply maponpaths. apply pair_2cell_pr1. } rewrite <- !lwhisker_vcomp. apply idpath. } rewrite !vassocl. apply maponpaths. refine (!_). etrans. { do 3 apply maponpaths. rewrite !vassocr. rewrite vcomp_linv. rewrite id2_left. apply idpath. } refine (!_). etrans. { do 3 apply maponpaths. apply binprod_ump_2cell_pr1. } etrans. { do 2 apply maponpaths. rewrite !vassocr. rewrite lassociator_rassociator. rewrite id2_left. apply idpath. } etrans. { apply maponpaths. rewrite !vassocr. rewrite lwhisker_vcomp. rewrite vcomp_linv. rewrite lwhisker_id2. rewrite id2_left. apply idpath. } rewrite !vassocr. rewrite lwhisker_lwhisker. rewrite !vassocl. apply maponpaths. rewrite !vassocr. rewrite <- vcomp_whisker. rewrite !vassocl. apply maponpaths. rewrite !vassocr. apply maponpaths_2. rewrite lwhisker_lwhisker_rassociator. apply idpath. - rewrite <- !rwhisker_vcomp. etrans. { apply maponpaths. apply pair_2cell_pr2. } etrans. { apply maponpaths_2. apply binprod_ump_2cell_pr2. } rewrite !vassocl. use vcomp_move_R_pM ; [ is_iso | ] ; cbn. refine (!_). etrans. { rewrite !vassocr. rewrite <- rwhisker_lwhisker. rewrite !vassocl. apply maponpaths_2. etrans. { apply maponpaths. apply pair_2cell_pr2. } rewrite <- !lwhisker_vcomp. apply idpath. } rewrite !vassocl. apply maponpaths. refine (!_). etrans. { do 3 apply maponpaths. rewrite !vassocr. rewrite vcomp_linv. rewrite id2_left. apply idpath. } refine (!_). etrans. { do 3 apply maponpaths. apply binprod_ump_2cell_pr2. } etrans. { do 2 apply maponpaths. rewrite !vassocr. rewrite lassociator_rassociator. rewrite id2_left. apply idpath. } etrans. { apply maponpaths. rewrite !vassocr. rewrite lwhisker_vcomp. rewrite vcomp_linv. rewrite lwhisker_id2. rewrite id2_left. apply idpath. } rewrite !vassocr. rewrite lwhisker_lwhisker. rewrite !vassocl. apply maponpaths. rewrite !vassocr. rewrite <- vcomp_whisker. rewrite !vassocl. apply maponpaths. rewrite !vassocr. apply maponpaths_2. rewrite lwhisker_lwhisker_rassociator. apply idpath. 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