Lemma qF (a0 : C) : ∏ (t t' : ∑ a : C, unitary dagD (H a) (H a0)) (f : pr1 t --> pr1 t'), #H f· pr2 t' = pr2 t -> #F f· #F (fH^-1 (pr2 t')) = #F (fH^-1 (pr2 t)). Proof. simpl. intros [a h] [a' h'] f L. simpl in L; simpl. rewrite <- functor_comp. apply maponpaths. apply (invmaponpathsweq (weq_from_fully_faithful fH a a0) (f· fH^-1 h') (fH^-1 h) ). inv_functor fH a a0. rewrite functor_comp. inv_functor fH a' a0. apply L. 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.whiskering. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.TotalCategoryFacts. Require Import UniMath.CategoryTheory.precomp_fully_faithful. Require Import UniMath.CategoryTheory.DaggerCategories.Categories. Require Import UniMath.CategoryTheory.DaggerCategories.Functors. Require Import UniMath.CategoryTheory.DaggerCategories.Transformations. Require Import UniMath.CategoryTheory.DaggerCategories.Unitary. Require Import UniMath.CategoryTheory.DaggerCategories.Univalence. Require Import UniMath.CategoryTheory.DaggerCategories.FunctorCategory. Require Import UniMath.CategoryTheory.DaggerCategories.Functors.FullyFaithful. Require Import UniMath.CategoryTheory.DaggerCategories.Functors.WeakEquivalences.