Definition arrow_category_equiv_comma_category_unit_data : nat_trans_data (functor_identity (arrow_category C)) (arrow_category_to_comma_category ∙ comma_category_to_arrow_category). 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.