Definition slice_counit_fun (f : slice X) : (functor_composite_data slice_to_fam_data fam_to_slice_data) f --> (functor_identity_data _) f. Proof. exists (fun h : (∑ x : X, hfiber (pr2 f) x) => pr1 (pr2 h)). simpl. apply funextsec. intro p. exact (!pr2 (pr2 p)). Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.CategoryTheory.Categories.StandardCategories.