/-- `(P ⟶ P⁺)⁺ = P⁺ ⟶ P⁺⁺` -/
@[simp]
theorem plusMap_toPlus : J.plusMap (J.toPlus P) = J.toPlus (J.plusObj P) := by
  ext X : 2
  refine' colimit.hom_ext (fun S => _)
  dsimp only [plusMap, toPlus]
  let e : S.unop ⟶ ⊤ := homOfLE (OrderTop.le_top _)
  rw [ι_colimMap, ← colimit.w _ e.op, ← Category.assoc, ← Category.assoc]
  congr 1
  refine' Multiequalizer.hom_ext _ _ _ (fun I => _)
  erw [Multiequalizer.lift_ι]
  simp only [unop_op, op_unop, diagram_map, Category.assoc, limit.lift_π,
    Multifork.ofι_π_app]
  let ee : (J.pullback (I.map e).f).obj S.unop ⟶ ⊤ := homOfLE (OrderTop.le_top _)
  erw [← colimit.w _ ee.op, ι_colimMap_assoc, colimit.ι_pre, diagramPullback_app,
    ← Category.assoc, ← Category.assoc]
  congr 1
  refine' Multiequalizer.hom_ext _ _ _ (fun II => _)
  convert (Multiequalizer.condition (S.unop.index P)
      ⟨_, _, _, II.f, 𝟙 _, I.f, II.f ≫ I.f, I.hf,
        Sieve.downward_closed _ I.hf _, by simp⟩) using 1
  · dsimp [diagram]
    cases I
    simp only [Category.assoc, limit.lift_π, Multifork.ofι_pt, Multifork.ofι_π_app,
      Cover.Arrow.map_Y, Cover.Arrow.map_f]
    rfl
  · erw [Multiequalizer.lift_ι]
    dsimp [Cover.index]
    

Proof state:
case h.e'_3.h
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{max v u, w} D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
P : Cᵒᵖ ⥤ D
inst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
X : Cᵒᵖ
S : (Cover J X.unop)ᵒᵖ
e : S.unop ⟶ ⊤ := homOfLE (_ : S.unop ≤ ⊤)
I : (Cover.index S.unop (plusObj J P)).L
ee : (pullback J (Cover.Arrow.map I e).f).obj S.unop ⟶ ⊤ :=
  homOfLE (_ : (pullback J (Cover.Arrow.map I e).f).obj S.unop ≤ ⊤)
II : (Cover.index (op ((pullback J (Cover.Arrow.map I e).f).obj S.unop)).unop P).L
e_1✝ :
  ((diagram J P X.unop).obj S ⟶
      MulticospanIndex.left (Cover.index (op ((pullback J (Cover.Arrow.map I e).f).obj S.unop)).unop P) II) =
    (multiequalizer (Cover.index S.unop P) ⟶
      MulticospanIndex.right (Cover.index S.unop P)
        { Y₁ := I.Y, Y₂ := II.Y, Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, f₁ := I.f, f₂ := II.f ≫ I.f,
          h₁ := (_ : (Cover.sieve S.unop).arrows I.f), h₂ := (_ : (Cover.sieve S.unop).arrows (II.f ≫ I.f)),
          w := (_ : II.f ≫ I.f = 𝟙 II.Y ≫ II.f ≫ I.f) })
⊢ Multiequalizer.ι
      { L := Cover.Arrow S.unop, R := Cover.Relation S.unop, fstTo := fun I => Cover.Relation.fst I,
        sndTo := fun I => Cover.Relation.snd I, left := fun I => P.obj (op I.Y), right := fun I => P.obj (op I.Z),
        fst := fun I => P.map I.g₁.op, snd := fun I => P.map I.g₂.op }
      (Cover.Arrow.base II) =
    Multiequalizer.ι
        { L := Cover.Arrow S.unop, R := Cover.Relation S.unop, fstTo := fun I => Cover.Relation.fst I,
          sndTo := fun I => Cover.Relation.snd I, left := fun I => P.obj (op I.Y), right := fun I => P.obj (op I.Z),
          fst := fun I => P.map I.g₁.op, snd := fun I => P.map I.g₂.op }
        (Cover.Relation.snd
          { Y₁ := I.Y, Y₂ := II.Y, Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, f₁ := I.f, f₂ := II.f ≫ I.f,
            h₁ := (_ : (Cover.sieve S.unop).arrows I.f), h₂ := (_ : (Cover.sieve S.unop).arrows (II.f ≫ I.f)),
            w := (_ : II.f ≫ I.f = 𝟙 II.Y ≫ II.f ≫ I.f) }) ≫
      P.map (𝟙 (op II.Y))

Next tactic:
simp only [Functor.map_id, Category.comp_id]