Lemma X_aux_type_center_of_contr_proof (b : D) (anot : C) (hnot : unitary dagD (H anot) b) : ∏ (t t' : ∑ a : C, unitary dagD (H a) b) (f : pr1 t --> pr1 t'), #H f· pr12 t' = pr12 t -> #F f· #F (fH^-1 (pr12 t'· dagD _ _ (pr1 hnot))) = #F (fH^-1 (pr12 t· dagD _ _ (pr1 hnot))). Proof. intros t t' f. destruct t as [a h]. destruct t' as [a' h']. simpl in *. intro star. rewrite <- functor_comp. apply maponpaths. apply (invmaponpathsweq (weq_from_fully_faithful fH a anot)). simpl. rewrite functor_comp. inv_functor fH a' anot. rewrite assoc. inv_functor fH a anot. rewrite <- star. 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.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.