proposition: Lemma AlgebrasForCoproductIntoProductcategory_is_functor : is_functor AlgebrasForCoproductIntoProductcategory_data.
proof: Proof. split; red. - intro X. apply funextsec. intro i. transparent assert (xx : (ob (total_precategory (algebra_disp_cat (F i))))). { exists (pr1 X). exact (taui_from_alg X i). } apply (mor_eq_total_category_when_locally_prop(D:=algebra_disp_cat (F i)) (is_locally_propositional_algebra_disp_cat (F i)) (xx:=xx)(yy:=xx)). apply idpath. - intros X Y Z f g. apply funextsec. intro i. transparent assert (xx : (ob (total_precategory (algebra_disp_cat (F i))))). { exists (pr1 X). exact (taui_from_alg X i). } transparent assert (zz : (ob (total_precategory (algebra_disp_cat (F i))))). { exists (pr1 Z). exact (taui_from_alg Z i). } apply (mor_eq_total_category_when_locally_prop(D:=algebra_disp_cat (F i)) (is_locally_propositional_algebra_disp_cat (F i)) (xx:=xx)(yy:=zz)). apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.NNO UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.ProductCategory UniMath.CategoryTheory.Limits.Coproducts