Lemma Precomp_of_dagger_weak_equiv_is_dagger_weak_equiv {C D E : category} {dagC : dagger C} {dagD : dagger D} {H : functor C D} {dagH : is_dagger_functor dagC dagD H} (wH : is_weak_dagger_equiv dagH) {dagE : dagger E} (univE : is_univalent_dagger dagE) : is_weak_dagger_equiv (dagger_pre_composition_functor_is_dagger_functor dagH dagE). Proof. split. - exact (Precomp_is_unitarily_eso _ wH univE). - exact (Precomp_is_ff _ wH). 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.