/-- The isomorphism between `P⁺ ⋙ F` and `(P ⋙ F)⁺`. -/
def plusCompIso : J.plusObj P ⋙ F ≅ J.plusObj (P ⋙ F) :=
  NatIso.ofComponents
    (fun X => by
      refine' _ ≪≫ HasColimit.isoOfNatIso (J.diagramCompIso F P X.unop)
      refine'
        (isColimitOfPreserves F
              (colimit.isColimit (J.diagram P (unop X)))).coconePointUniqueUpToIso
          (colimit.isColimit _))
    (by
      intro X Y f
      apply (isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).hom_ext
      intro W
      dsimp [plusObj, plusMap]
      simp only [Functor.map_comp, Category.assoc]
      slice_rhs 1 2 =>
        erw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).fac]
      slice_lhs 1 3 =>
        simp only [← F.map_comp]
        dsimp [colimMap, IsColimit.map, colimit.pre]
        simp only [colimit.ι_desc_assoc, colimit.ι_desc]
        dsimp [Cocones.precompose]
        simp only [Category.assoc, colimit.ι_desc]
        dsimp [Cocone.whisker]
        rw [F.map_comp]
      simp only [Category.assoc]
      slice_lhs 2 3 =>
        erw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P Y.unop))).fac]
      dsimp
      simp only [HasColimit.isoOfNatIso_ι_hom_assoc, GrothendieckTopology.diagramPullback_app,
        colimit.ι_pre, HasColimit.isoOfNatIso_ι_hom, ι_colimMap_assoc]
      simp only [← Category.assoc]
      dsimp
      

Proof state:
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P : Cᵒᵖ ⥤ D
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
X Y : Cᵒᵖ
f : X ⟶ Y
W : (Cover J X.unop)ᵒᵖ
⊢ (F.map
          (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))
            (fun I => Multiequalizer.ι (Cover.index W.unop P) (Cover.Arrow.base I))
            (_ :
              ∀ (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),
                Multiequalizer.ι (Cover.index W.unop P)
                      (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) ≫
                    MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =
                  Multiequalizer.ι (Cover.index W.unop P)
                      (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) ≫
                    MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) ≫
        (diagramCompIso J F P Y.unop).hom.app (op (Cover.pullback W.unop f.unop))) ≫
      colimit.ι (diagram J (P ⋙ F) Y.unop) (op (Cover.pullback W.unop f.unop)) =
    ((diagramCompIso J F P X.unop).hom.app W ≫
        Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) (P ⋙ F))
          (multiequalizer (Cover.index W.unop (P ⋙ F)))
          (fun I => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) (Cover.Arrow.base I))
          (_ :
            ∀ (I : (Cover.index ((pullback J f.unop).op.obj W).unop (P ⋙ F)).R),
              Multiequalizer.ι (Cover.index W.unop (P ⋙ F))
                    (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ F)) (Cover.Relation.base I)) ≫
                  MulticospanIndex.fst (Cover.index W.unop (P ⋙ F)) (Cover.Relation.base I) =
                Multiequalizer.ι (Cover.index W.unop (P ⋙ F))
                    (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ F)) (Cover.Relation.base I)) ≫
                  MulticospanIndex.snd (Cover.index W.unop (P ⋙ F)) (Cover.Relation.base I))) ≫
      colimit.ι (diagram J (P ⋙ F) Y.unop) (op (Cover.pullback W.unop f.unop))

Next tactic:
congr 1