proposition: Lemma make_algebraic_theory_z_iso_inv_comp_ax : ∏ m n f g, mor_comp_ax make_algebraic_theory_z_iso_inv_data m n f g.
proof: Proof. intros m n f g. refine (!_ @ maponpaths (λ x, x _) (z_iso_inv_after_z_iso (F n))). refine (maponpaths (λ x, inv_from_z_iso (F n) x) (Hcomp _ _ _ _) @ _). apply maponpaths. refine (maponpaths (λ x, (x f) • _) (z_iso_after_z_iso_inv (F m)) @ _). apply maponpaths. apply funextfun. intro. exact (maponpaths (λ x, x (g _)) (z_iso_after_z_iso_inv (F n))). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Limits UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.AlgebraicTheories.IndexedSetCategory UniMath.AlgebraicTheories.AlgebraicTheoryCategoryCore UniMath.AlgebraicTheories.AlgebraicTheories UniMath.AlgebraicTheories.AlgebraicTheoryMorphisms