Lemma ψ_naturality (A B: C)(h: B --> A)(f: L A --> X): ψ B (#L h· f) = #L (#F h)· ψ A f. Proof. assert (ψ_is_nat := nat_trans_ax ψ); assert (ψ_is_nat_inst1 := ψ_is_nat _ _ h). assert (ψ_is_nat_inst2 := eqtohomot ψ_is_nat_inst1 f). apply ψ_is_nat_inst2. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.Adjunctions.Core.