Proposition transportf_psfunctor_into_cat {C : category} {F : psfunctor (cat_to_bicat C^op) bicat_of_univ_cats} {x y : C} (f : x --> y) (yy : pr1 (F y)) {g h : x --> y} (p : g = h) (ff : pr1 (# F f) yy --> pr1 (# F g) yy) : transportf (λ z, pr1 (#F f) yy --> pr1 (#F z) yy) p ff = ff · pr1 (##F p) yy. Proof. induction p ; cbn. refine (!(id_right _) @ _). apply maponpaths. refine (!(nat_trans_eq_pointwise (psfunctor_id2 F _) _) @ _). refine (maponpaths (λ z, pr1 (##F z) yy) _). apply homset_property. Qed.
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.IndexedCategories.IndexedCategory. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.OppositeCategory.Core. 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. Section PseudoFunctorFromCat. End PseudoFunctorFromCat.