proposition: Lemma invweqweq (F : algebraic_theory_morphism (free_theory A) T) : function_to_theory_morphism (theory_morphism_to_function F) = F.
proof: Proof. apply algebraic_theory_morphism_eq. intros ? f. induction f. - exact (!mor_pr _ _). - refine (lift_constant_eq _ _ _ _ (idpath _) @ _). exact (!mor_comp F _ _ : _ = F _ (lift_constant _ _)). 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