Definition precomp_prod_1cell {a b c₁ c₂ : B} (f : a --> b) (g₁ : b --> c₁) (g₂ : b --> c₂) : f · ⟨ g₁ , g₂ ⟩ ==> ⟨ f · g₁ , f · g₂ ⟩. Proof. use (binprod_ump_2cell (pr2 (binprod_of B _ _))). - exact (rassociator _ _ _ • (f ◃ prod_1cell_pr1 _ _) • (prod_1cell_pr1 _ _)^-1). - exact (rassociator _ _ _ • (f ◃ prod_1cell_pr2 _ _) • (prod_1cell_pr2 _ _)^-1). Defined.
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.