proposition: Proposition underlying_Coequalizer_arr_in {w : C} (h : y --> w) (q : f · h = g · h) : enriched_coequalizer_cocone_in a · underlying_Coequalizer_arr h q = h.
proof: Proof. unfold underlying_Coequalizer_arr, enriched_coequalizer_cocone_in. use (invmaponpathsweq (make_weq _ (isweq_enriched_from_arr E _ _))) ; cbn. rewrite enriched_from_arr_comp. rewrite !enriched_from_to_arr. refine (_ @ EqualizerCommutes (is_coequalizer_enriched_to_Equalizer Ha w) I_{V} (enriched_from_arr E h) _). rewrite tensor_split'. rewrite !assoc. rewrite mon_linvunitor_I_mon_rinvunitor_I. rewrite <- tensor_rinvunitor. cbn. unfold precomp_arr. rewrite !assoc. apply maponpaths_2. rewrite !assoc'. do 3 apply maponpaths. refine (!_). apply enriched_from_to_arr. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.EnrichedCats.Enrichment UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers