theorem toPlus_apply {X : C} {P : Cᵒᵖ ⥤ D} (S : J.Cover X) (x : Meq P S) (I : S.Arrow) :
    (J.toPlus P).app _ (x I) = (J.plusObj P).map I.f.op (mk x) := by
  dsimp only [toPlus, plusObj]
  delta Cover.toMultiequalizer
  dsimp [mk]
  erw [← comp_apply]
  rw [ι_colimMap_assoc, colimit.ι_pre, comp_apply, comp_apply]
  dsimp only [Functor.op]
  let e : (J.pullback I.f).obj (unop (op S)) ⟶ ⊤ := homOfLE (OrderTop.le_top _)
  rw [← colimit.w _ e.op]
  erw [comp_apply]
  apply congr_arg
  apply Concrete.multiequalizer_ext
  intro i
  dsimp [diagram]
  

Proof state:
case h.h
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
P : Cᵒᵖ ⥤ D
S : Cover J X
x : Meq P S
I : Cover.Arrow S
e : (pullback J I.f).obj (op S).unop ⟶ ⊤ := homOfLE (_ : (pullback J I.f).obj (op S).unop ≤ ⊤)
i : (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P).L
⊢ (Multiequalizer.ι (Cover.index (Cover.pullback S I.f) P) i)
      ((Multiequalizer.lift (Cover.index (Cover.pullback S I.f) P) (multiequalizer (Cover.index ⊤ P))
          (fun I_1 => Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.map I_1 (homOfLE (_ : Cover.pullback S I.f ≤ ⊤))))
          (_ :
            ∀ (I_1 : (Cover.index (op (Cover.pullback S I.f)).unop P).R),
              Multiequalizer.ι (Cover.index (op ⊤).unop P)
                    (MulticospanIndex.fstTo (Cover.index (op ⊤).unop P)
                      (Cover.Relation.map I_1 (homOfLE (_ : Cover.pullback S I.f ≤ ⊤)).op.unop)) ≫
                  MulticospanIndex.fst (Cover.index (op ⊤).unop P)
                    (Cover.Relation.map I_1 (homOfLE (_ : Cover.pullback S I.f ≤ ⊤)).op.unop) =
                Multiequalizer.ι (Cover.index (op ⊤).unop P)
                    (MulticospanIndex.sndTo (Cover.index (op ⊤).unop P)
                      (Cover.Relation.map I_1 (homOfLE (_ : Cover.pullback S I.f ≤ ⊤)).op.unop)) ≫
                  MulticospanIndex.snd (Cover.index (op ⊤).unop P)
                    (Cover.Relation.map I_1 (homOfLE (_ : Cover.pullback S I.f ≤ ⊤)).op.unop)))
        ((Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)
            (_ :
              ∀ (I_1 : (Cover.index ⊤ P).R),
                (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I_1) ≫
                    MulticospanIndex.fst (Cover.index ⊤ P) I_1 =
                  (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I_1) ≫
                    MulticospanIndex.snd (Cover.index ⊤ P) I_1))
          (↑x I))) =
    (Multiequalizer.ι (Cover.index (Cover.pullback S I.f) P) i)
      ((Multiequalizer.lift (Cover.index (Cover.pullback S I.f) P) (multiequalizer (Cover.index S P))
          (fun I_1 => Multiequalizer.ι (Cover.index S P) (Cover.Arrow.base I_1))
          (_ :
            ∀ (I_1 : (Cover.index ((pullback J I.f).op.obj (op S)).unop P).R),
              Multiequalizer.ι (Cover.index (op S).unop P)
                    (MulticospanIndex.fstTo (Cover.index (op S).unop P) (Cover.Relation.base I_1)) ≫
                  MulticospanIndex.fst (Cover.index (op S).unop P) (Cover.Relation.base I_1) =
                Multiequalizer.ι (Cover.index (op S).unop P)
                    (MulticospanIndex.sndTo (Cover.index (op S).unop P) (Cover.Relation.base I_1)) ≫
                  MulticospanIndex.snd (Cover.index (op S).unop P) (Cover.Relation.base I_1)))
        ((Meq.equiv P S).symm x))

Next tactic:
rw [← comp_apply, ← comp_apply, ← comp_apply, Multiequalizer.lift_ι, Multiequalizer.lift_ι,
    Multiequalizer.lift_ι]