Lemma is_nat_trans_ψ_from_comps : is_nat_trans ψ_source ψ_target (λ A (f : yoneda_objects_ob D X (L A)), θ A · # G f · ρ). Proof. intros A B h; apply funextfun; intro f; cbn. rewrite functor_comp, !assoc. assert (θ_nat_trans_ax := nat_trans_ax θ); simpl in θ_nat_trans_ax. now rewrite <- θ_nat_trans_ax. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Chains.Chains. Require Import UniMath.CategoryTheory.Chains.Adamek. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.whiskering.