proposition: Lemma colimσ_on_welldefined (A : C) (m : vertex nat_graph) : T ⊗^{ V}_{l} pair_diagram_horizontal_arrow (free_monoid_coeq_sequence_on A m) · next_pair_diagram (free_monoid_coeq_sequence_on A m) = free_monoid_coeq_sequence_on A m · pair_diagram_horizontal_arrow (next_pair_diagram (free_monoid_coeq_sequence_on A m)).
proof: Proof. set (coeq := next_pair_diagram_coeq (free_monoid_coeq_sequence_on A m)). set (coeq_comm := CoequalizerArrowEq _ coeq). etrans. exact (pathsinv0 coeq_comm). apply pathsinv0. etrans. apply assoc. reflexivity. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Limits.Graphs.Coequalizers UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Chains.Chains UniMath.Combinatorics.StandardFiniteSets UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids UniMath.ModelCategories.Generated.GenericFreeMonoid UniMath.ModelCategories.Generated.MonoidalHelpers UniMath.ModelCategories.Helpers