proposition: Lemma T_preserves_diagram_convergence_inverse_in_precat : is_inverse_in_precat (pair_diagram_horizontal_arrow (free_monoid_coeq_sequence_limit_ordinal_step_on A)) T_preserves_diagram_convergence_inv.
proof: Proof. split. - apply pathsinv0. use colim_endo_is_identity. intro u. etrans. apply assoc. etrans. apply cancel_postcomposition. apply assoc. etrans. apply assoc'. etrans. apply cancel_precomposition. apply CoequalizerArrowComm. etrans. apply cancel_postcomposition. { 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 assoc'. etrans. apply cancel_precomposition. exact (TcolimIn_T_preserves_diagram_coeqout_arrow_on u). etrans. apply assoc. exact (colimInCommutes (free_monoid_coeq_sequence_colim_on A) _ _ (idpath (S u))). - apply pathsinv0. use CoequalizerEndo_is_identity. etrans. apply assoc. etrans. apply cancel_postcomposition. apply CoequalizerArrowComm. etrans. apply assoc'. etrans. apply cancel_precomposition. apply assoc. etrans. apply cancel_precomposition. { etrans. apply cancel_postcomposition. apply assoc'. exact (pathsinv0 (CoequalizerArrowEq _ (free_monoid_coeq_sequence_limit_step_coeq A))). } apply (pre_comp_with_z_iso_is_inj iso). use colimArrowUnique'. intro u. etrans. apply cancel_precomposition. { etrans. apply assoc. etrans. apply cancel_postcomposition. exact (pr12 iso). apply id_left. } etrans. apply assoc. etrans. apply cancel_postcomposition. apply colimArrowCommutes. apply pathsinv0. etrans. apply assoc. etrans. apply cancel_postcomposition. apply (colimArrowCommutes (free_monoid_coeq_sequence_leftwhisker_colim_on T A)). etrans. apply assoc'. etrans. apply cancel_postcomposition. apply (bifunctor_rightid V). apply id_left. 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