proposition: Lemma T_preserves_diagram_coeqout_arrow_on_is_coeqout : can A · T_preserves_diagram_coeqout_arrow_on = top A · T_preserves_diagram_coeqout_arrow_on.
proof: Proof. use colimArrowUnique'. intro u. etrans. apply assoc. etrans. apply cancel_postcomposition. use (colimArrowCommutes ((free_monoid_coeq_sequence_leftwhisker_colim_on T A))). etrans. exact (TcolimIn_T_preserves_diagram_coeqout_arrow_on u). apply pathsinv0. etrans. apply assoc. etrans. apply cancel_postcomposition. { etrans. apply assoc. etrans. apply cancel_postcomposition. apply colimOfArrowsIn. etrans. apply assoc'. apply cancel_precomposition. etrans. apply assoc. etrans. apply cancel_postcomposition. exact (chain_shift_left_colimIn_commutes u). etrans. apply assoc. etrans. apply cancel_postcomposition. exact (pathsinv0 (monoidal_leftunitorinvnat V _ _ _)). etrans. apply assoc'. apply cancel_precomposition. exact (pathsinv0 (whiskerscommutes V (bifunctor_equalwhiskers V) _ _)). } etrans. apply cancel_postcomposition. apply assoc. etrans. apply cancel_postcomposition. apply assoc. etrans. apply assoc'. etrans. apply cancel_precomposition. exact (TcolimIn_T_preserves_diagram_coeqout_arrow_on (S u)). etrans. apply assoc'. etrans. apply assoc'. apply cancel_precomposition. etrans. apply assoc. etrans. apply assoc. exact ( colimInCommutes (free_monoid_coeq_sequence_colim_on A) _ _ (idpath (S (S u)))). 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