proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.TotalCategoryFacts UniMath.CategoryTheory.precomp_fully_faithful UniMath.CategoryTheory.DaggerCategories.Categories UniMath.CategoryTheory.DaggerCategories.Functors UniMath.CategoryTheory.DaggerCategories.Transformations UniMath.CategoryTheory.DaggerCategories.Unitary UniMath.CategoryTheory.DaggerCategories.Univalence UniMath.CategoryTheory.DaggerCategories.FunctorCategory UniMath.CategoryTheory.DaggerCategories.Functors.FullyFaithful UniMath.CategoryTheory.DaggerCategories.Functors.WeakEquivalences