proposition: Lemma lifting_coprod_lp_impl_lifting_data : filler (arrow_mor_comm (morcls_lp_coprod_diagram)) -> right_lifting_data J f.
proof: Proof. intro l. intros g Jg h k H. destruct l as [ltot [Hl1 Hl2]]. simpl in Hl1, Hl2, ltot. set (Jtot := total_category (morcls_disp J)). set (g' := (g,, Jg) : Jtot). set (Hlp := (g',, (make_dirprod h k,, H)) : morcls_lp J f). set (codg_in := (CoproductIn (morcls_lp_cod_coprod) Hlp)). set (domg_in := (CoproductIn (morcls_lp_dom_coprod) Hlp)). exists (codg_in · ltot). split. - assert (h = domg_in · (arrow_mor00 (morcls_lp_coprod_diagram))) as Hh. { apply pathsinv0. exact (CoproductInCommutes (morcls_lp_dom_coprod) _ Hlp). } apply pathsinv0. etrans. exact Hh. apply pathsinv0. assert (g · codg_in = domg_in · (morcls_lp_coprod)) as Hg. { unfold codg_in, domg_in, morcls_lp_coprod. rewrite (CoproductOfArrows'In _ _ (morcls_lp_dom_coprod)). reflexivity. } etrans. apply assoc. etrans. apply maponpaths_2. exact Hg. etrans. apply assoc'. apply cancel_precomposition. exact Hl1. - etrans. apply assoc'. etrans. apply maponpaths. exact Hl2. apply pathsinv0. apply pathsinv0. exact (CoproductInCommutes (morcls_lp_cod_coprod) _ Hlp). Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monads.Monads UniMath.CategoryTheory.Monads.MonadAlgebras UniMath.CategoryTheory.catiso UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.Coequalizers UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.SIP UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Equivalences UniMath.CategoryTheory.DisplayedCats.Examples.Opposite UniMath.CategoryTheory.DisplayedCats.Examples.Arrow UniMath.CategoryTheory.DisplayedCats.Examples.Three UniMath.ModelCategories.MorphismClass UniMath.ModelCategories.Retract UniMath.ModelCategories.Lifting UniMath.ModelCategories.NWFS UniMath.ModelCategories.NWFSisWFS UniMath.ModelCategories.Helpers