proposition: Lemma is_functional_unit_element : is_functional unit_element 1.
proof: Proof. unfold unit_element, is_functional, make_functional. cbn -[weqvecfun action]. rewrite (move_action_through_vector_1 _ _). rewrite <- comp_action. cbn -[weqvecfun action]. rewrite subst_abs. rewrite subst_app. do 2 rewrite subst_var. do 2 extend_tuple_2. cbn. rewrite inflate_abs. rewrite beta_equality. rewrite subst_var. cbn. now rewrite subst_var. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.Algebra.Monoids UniMath.Combinatorics.Tuples UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Presheaf UniMath.Combinatorics.StandardFiniteSets UniMath.Combinatorics.Vectors UniMath.AlgebraicTheories.AlgebraicTheories UniMath.AlgebraicTheories.Algebras UniMath.AlgebraicTheories.Examples.LambdaCalculus UniMath.AlgebraicTheories.LambdaTheories UniMath.AlgebraicTheories.LambdaCalculus UniMath.AlgebraicTheories.LambdaTheoryCategory