proposition: Proposition enriched_conical_colim_from_coprod_coequalizers_unique : isaprop (∑ (φ : v --> E ⦃ colim , w ⦄), ∏ (i : I), φ · precomp_arr E w (enriched_conical_colim_in_from_coprod_coequalizers i) = fs i).
proof: Proof. use invproofirrelevance. intros φ₁ φ₂. use subtypePath. { intro. use impred ; intro. apply homset_property. } use (EqualizerInsEq (is_coequalizer_enriched_to_Equalizer _ _ _ enriched_conical_colim_ob_from_coprod_coequalizers_is_coequalizer w)). use (ProductArrow_eq _ _ _ (is_coprod_enriched_to_Product _ _ is_coprod_tar w)). intro i ; cbn. rewrite !assoc'. rewrite <- !precomp_arr_comp. exact (pr2 φ₁ i @ !(pr2 φ₂ i)). 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.OppositeCategory.Core UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Products UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.EnrichedCats.Enrichment UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCoproducts UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCoequalizers