Definition pstrans_from_cat_into_cat_laws {C : category} {F G : psfunctor (cat_to_bicat C) bicat_of_univ_cats} (τ : pstrans_from_cat_into_cat_data F G) : UU := (∏ (x : C) (xx : pr1 (F x)), pr11 (psfunctor_id G x) (pr1 (τ x) xx) · pr1 (pstrans_from_cat_into_cat_data_nat τ (id₁ x)) xx = # (pr1 (τ x)) (pr11 (psfunctor_id F x) xx)) × (∏ (x y z : C) (f : x --> y) (g : y --> z) (xx : pr1 (F x)), pr11 (psfunctor_comp G f g) (pr1 (τ x) xx) · pr1 (pstrans_from_cat_into_cat_data_nat τ (f · g)) xx = # (pr1 (# G g)) ((pr11 (pstrans_from_cat_into_cat_data_nat τ f)) xx) · (pr11 (pstrans_from_cat_into_cat_data_nat τ g)) (pr1 (# F f) xx) · # (pr1 (τ z)) (pr11 (psfunctor_comp F f g) xx)).
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedCategory. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedFunctor. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Discreteness. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Examples.DiscreteBicat. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.PseudoFunctorsIntoCat. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Section PseudoTransFromCat. End PseudoTransFromCat.