proposition: Lemma free_functor_is_functor : is_functor free_functor_data.
proof: Proof. split. - intro A. use (algebraic_theory_morphism_eq (T := free_theory A) (T' := free_theory A)). intros n f. now induction f. - intros a b c d e. use (algebraic_theory_morphism_eq (T := free_theory a) (T' := free_theory c)). intros n f. now induction f. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.Limits UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.coslicecat UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.Combinatorics.StandardFiniteSets UniMath.Combinatorics.Vectors UniMath.AlgebraicTheories.AlgebraCategory UniMath.AlgebraicTheories.AlgebraCategoryCore UniMath.AlgebraicTheories.AlgebraicTheories UniMath.AlgebraicTheories.AlgebraicTheoryCategoryCore UniMath.AlgebraicTheories.AlgebraicTheoryMorphisms UniMath.AlgebraicTheories.AlgebraMorphisms UniMath.AlgebraicTheories.Algebras