Lemma covyoneda_map_2_ax (C : category) (c : C) (F : functor C HSET) (x : pr1 (F c)) : is_nat_trans (pr1 (covyoneda C c)) F (fun (d : C) (f : C⟦c, d⟧) => #F f x). Proof. intros a b f; simpl in *. apply funextsec. unfold covyoneda_objects_ob; intro g. set (H:= @functor_comp _ _ F _ _ b g). unfold functor_comp in H; unfold opp_precat_data in H; simpl in *. apply (toforallpaths _ _ _ (H f) x). Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.whiskering.