proposition: Proposition coequalizer_enriched_from_underlying_map_inv₁ (h : I_{V} --> E ⦃ a , w ⦄) : coequalizer_enriched_from_underlying_map (h · coequalizer_enriched_to_equalizer EqV a w) = h.
proof: Proof. unfold coequalizer_enriched_from_underlying_map. refine (_ @ enriched_from_to_arr E h). apply maponpaths. use (isCoequalizerOutsEq eq_a). etrans. { apply (CoequalizerCommutes (make_Coequalizer _ _ _ _ eq_a)). } unfold enriched_coequalizer_cocone_in. rewrite (enriched_to_arr_comp E). apply maponpaths. rewrite tensor_split'. rewrite !assoc. rewrite mon_linvunitor_I_mon_rinvunitor_I. rewrite <- tensor_rinvunitor. rewrite enriched_from_to_arr. rewrite !assoc'. apply maponpaths. unfold coequalizer_enriched_to_equalizer. rewrite EqualizerCommutes. rewrite !assoc. apply idpath. 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