Lemma unitary_functors_eq_from_unitary_functors_naturality (α : unitary_functors dagF dagG) : ∏ (x y : C) (f : C ⟦ x, y ⟧), # G f = dagD (F x) (G x) ((pr1 α) x) · # F f · (pr1 α) y. Proof. intros c1 c2 f. apply pathsinv0. refine (assoc' _ _ _ @ _). refine (maponpaths (compose _) (pr21 α _ _ _) @ _). refine (assoc _ _ _ @ _). refine (_ @ id_left _). apply maponpaths_2, (pr2 α). 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.FunctorCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DaggerCategories.Categories. Require Import UniMath.CategoryTheory.DaggerCategories.Unitary. Require Import UniMath.CategoryTheory.DaggerCategories.Univalence. Require Import UniMath.CategoryTheory.DaggerCategories.Functors. Require Import UniMath.CategoryTheory.DaggerCategories.Transformations.