Lemma binprod_rwhisker {a₁ a₂ a₃ b₁ b₂ b₃ : B} (f₁ g₁ : a₁ --> a₂) (f₂ g₂ : b₁ --> b₂) {h₁ : a₂ --> a₃} {h₂ : b₂ --> b₃} (τ₁ : f₁ ==> g₁) (τ₂ : f₂ ==> g₂) : pair_1cell_comp f₁ h₁ f₂ h₂ • ((τ₁ ▹ h₁) ⊗₂ (τ₂ ▹ h₂)) = ((τ₁ ⊗₂ τ₂) ▹ (h₁ ⊗₁ h₂)) • pair_1cell_comp g₁ h₁ g₂ h₂. 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. refine (!_). etrans. { apply maponpaths. apply binprod_ump_2cell_pr1. } etrans. { rewrite !vassocr. rewrite rwhisker_rwhisker_alt. rewrite !vassocl. apply idpath. } apply maponpaths. etrans. { rewrite !vassocr. rewrite vcomp_whisker. rewrite !vassocl. apply idpath. } apply maponpaths. etrans. { rewrite !vassocr. rewrite <- rwhisker_rwhisker. rewrite !vassocl. apply idpath. } apply maponpaths. etrans. { apply maponpaths_2. apply maponpaths. apply binprod_ump_2cell_pr1. } rewrite <- !rwhisker_vcomp. rewrite !vassocl. apply maponpaths. etrans. { apply maponpaths. rewrite !vassocr. rewrite rwhisker_vcomp. rewrite vcomp_linv. rewrite id2_rwhisker. rewrite id2_left. apply idpath. } etrans. { rewrite !vassocr. rewrite <- rwhisker_lwhisker_rassociator. rewrite !vassocl. apply idpath. } apply maponpaths. rewrite !vassocr. rewrite vcomp_linv. rewrite id2_left. apply idpath. - rewrite <- !rwhisker_vcomp. etrans. { apply maponpaths. apply pair_2cell_pr2. } etrans. { apply maponpaths_2. apply binprod_ump_2cell_pr2. } rewrite !vassocl. refine (!_). etrans. { apply maponpaths. apply binprod_ump_2cell_pr2. } etrans. { rewrite !vassocr. rewrite rwhisker_rwhisker_alt. rewrite !vassocl. apply idpath. } apply maponpaths. etrans. { rewrite !vassocr. rewrite vcomp_whisker. rewrite !vassocl. apply idpath. } apply maponpaths. etrans. { rewrite !vassocr. rewrite <- rwhisker_rwhisker. rewrite !vassocl. apply idpath. } apply maponpaths. etrans. { apply maponpaths_2. apply maponpaths. apply binprod_ump_2cell_pr2. } rewrite <- !rwhisker_vcomp. rewrite !vassocl. apply maponpaths. etrans. { apply maponpaths. rewrite !vassocr. rewrite rwhisker_vcomp. rewrite vcomp_linv. rewrite id2_rwhisker. rewrite id2_left. apply idpath. } etrans. { rewrite !vassocr. rewrite <- rwhisker_lwhisker_rassociator. rewrite !vassocl. apply idpath. } apply maponpaths. rewrite !vassocr. rewrite vcomp_linv. rewrite id2_left. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Univalence.