-- Module: Mathlib.CategoryTheory.Sites.CompatiblePlus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v_1, v_2, u_1, u_2} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_1, u, u_1} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ D
inst_6 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_7 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst_8 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X Y : Cᵒᵖ
f : X ⟶ Y
W : (J.Cover (unop X))ᵒᵖ
simp :
  Eq (α := F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y)))
    (CategoryStruct.comp (Y := F.obj (multiequalizer (((unop W).pullback f.unop).index P)))
        (F.map
            (Multiequalizer.lift (((unop W).pullback f.unop).index P) (multiequalizer ((unop W).index P))
              (fun (I : ((unop W).pullback f.unop).Arrow) ↦ Multiequalizer.ι ((unop W).index P) I.base) ⋯) :
          F.obj (multiequalizer ((unop W).index P)) ⟶ F.obj (multiequalizer (((unop W).pullback f.unop).index P)))
        (CategoryStruct.comp (Y := obj (J.diagram (P ⋙ F) (unop Y)) (op ((unop W).pullback f.unop)))
            (NatTrans.app (Iso.hom (J.diagramCompIso F P (unop Y))) (op ((unop W).pullback f.unop)) :
              obj (J.diagram P (unop Y) ⋙ F) (op ((unop W).pullback f.unop)) ⟶
                obj (J.diagram (P ⋙ F) (unop Y)) (op ((unop W).pullback f.unop)))
            (colimit.ι (J.diagram (P ⋙ F) (unop Y)) (op ((unop W).pullback f.unop)) :
              obj (J.diagram (P ⋙ F) (unop Y)) (op ((unop W).pullback f.unop)) ⟶
                colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y))) :
          obj (J.diagram P (unop Y) ⋙ F) (op ((unop W).pullback f.unop)) ⟶
            colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y))) :
      F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y)))
    (CategoryStruct.comp (Y := obj (J.diagram (P ⋙ F) (unop X)) W)
        (NatTrans.app (Iso.hom (J.diagramCompIso F P (unop X))) W :
          obj (J.diagram P (unop X) ⋙ F) W ⟶ obj (J.diagram (P ⋙ F) (unop X)) W)
        (CategoryStruct.comp (Y := ((J.pullback f.unop).op ⋙ J.diagram (P ⋙ F) (unop Y)).obj W)
            (Multiequalizer.lift ((unop ((J.pullback f.unop).op.obj W)).index (P ⋙ F))
              (obj (J.diagram (P ⋙ F) (unop X)) W)
              (fun (I : (unop ((J.pullback f.unop).op.obj W)).shape.L) ↦
                Multiequalizer.ι ((unop W).index (P ⋙ F)) (Cover.Arrow.base I))
              ⋯)
            (colimit.ι (J.diagram (P ⋙ F) (unop Y)) ((J.pullback f.unop).op.obj W)) :
          obj (J.diagram (P ⋙ F) (unop X)) W ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y))) :
      obj (J.diagram P (unop X) ⋙ F) W ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y)))
⊢ Eq (α := F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y)))
    (CategoryStruct.comp (Y := F.obj (multiequalizer (((unop W).pullback f.unop).index P)))
        (F.map
            (Multiequalizer.lift (((unop W).pullback f.unop).index P) (multiequalizer ((unop W).index P))
              (fun (I : ((unop W).pullback f.unop).Arrow) ↦ Multiequalizer.ι ((unop W).index P) I.base) ⋯) :
          F.obj (multiequalizer ((unop W).index P)) ⟶ F.obj (multiequalizer (((unop W).pullback f.unop).index P)))
        (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram P (unop Y) ⋙ F))
            (colimit.ι (J.diagram P (unop Y) ⋙ F) (op ((unop W).pullback f.unop)) :
              obj (J.diagram P (unop Y) ⋙ F) (op ((unop W).pullback f.unop)) ⟶
                colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram P (unop Y) ⋙ F))
            (Iso.hom (HasColimit.isoOfNatIso (J.diagramCompIso F P (unop Y))) :
              colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram P (unop Y) ⋙ F) ⟶
                colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y))) :
          F.obj (multiequalizer (((unop W).pullback f.unop).index P)) ⟶
            colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y))) :
      F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y)))
    (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F))
        (colimit.ι (J.diagram P (unop X) ⋙ F) W :
          obj (J.diagram P (unop X) ⋙ F) W ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F))
        (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (P ⋙ F) (unop X)))
            (Iso.hom (HasColimit.isoOfNatIso (J.diagramCompIso F P (unop X))) :
              colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F) ⟶
                colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (P ⋙ F) (unop X)))
            (CategoryStruct.comp (Y := colimit ((J.pullback f.unop).op ⋙ J.diagram (P ⋙ F) (unop Y)))
                (colimMap (J.diagramPullback (P ⋙ F) f.unop))
                (colimit.pre (J.diagram (P ⋙ F) (unop Y)) (J.pullback f.unop).op) :
              colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (P ⋙ F) (unop X)) ⟶
                colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y))) :
          colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F) ⟶
            colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y))) :
      F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y)))

Tactic:
simp only [HasColimit.isoOfNatIso_ι_hom_assoc, GrothendieckTopology.diagramPullback_app,
        colimit.ι_pre, HasColimit.isoOfNatIso_ι_hom, ι_colimMap_assoc]

Premises used: CategoryTheory.Limits.ι_colimMap_assoc, CategoryTheory.Limits.colimit.ι_pre, CategoryTheory.GrothendieckTopology.diagramPullback_app, CategoryTheory.Limits.HasColimit.isoOfNatIso_ι_hom_assoc, CategoryTheory.Limits.HasColimit.isoOfNatIso_ι_hom