Definition arrow_category_equiv_comma_category_counit_data : nat_trans_data (comma_category_to_arrow_category ∙ arrow_category_to_comma_category) (functor_identity (comma (functor_identity C) (functor_identity C))). Proof. cbn. refine (λ x, (identity _ ,, identity _) ,, _). abstract (cbn ; rewrite id_left, id_right ; apply idpath). Defined.
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.Core.Isos. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.CommaCategories. Require Import UniMath.CategoryTheory.Adjunctions.Core.