Theorem: CategoryTheory.GrothendieckTopology.plusCompIso_whiskerLeft
File: Mathlib/CategoryTheory/Sites/CompatiblePlus.lean

tactic: ext X
state_before: C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
⊢ whiskerLeft (J.plusObj P) η ≫ (J.plusCompIso G P).hom = (J.plusCompIso F P).hom ≫ J.plusMap (whiskerLeft P η)
state_after: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
⊢ (whiskerLeft (J.plusObj P) η ≫ (J.plusCompIso G P).hom).app X =
    ((J.plusCompIso F P).hom ≫ J.plusMap (whiskerLeft P η)).app X

tactic: apply (isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).hom_ext
state_before: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
⊢ (whiskerLeft (J.plusObj P) η ≫ (J.plusCompIso G P).hom).app X =
    ((J.plusCompIso F P).hom ≫ J.plusMap (whiskerLeft P η)).app X
state_after: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
⊢ ∀ (j : (J.Cover X.unop)ᵒᵖ),
    (F.mapCocone (colimit.cocone (J.diagram P X.unop))).ι.app j ≫
        (whiskerLeft (J.plusObj P) η ≫ (J.plusCompIso G P).hom).app X =
      (F.mapCocone (colimit.cocone (J.diagram P X.unop))).ι.app j ≫
        ((J.plusCompIso F P).hom ≫ J.plusMap (whiskerLeft P η)).app X

tactic: intro W
state_before: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
⊢ ∀ (j : (J.Cover X.unop)ᵒᵖ),
    (F.mapCocone (colimit.cocone (J.diagram P X.unop))).ι.app j ≫
        (whiskerLeft (J.plusObj P) η ≫ (J.plusCompIso G P).hom).app X =
      (F.mapCocone (colimit.cocone (J.diagram P X.unop))).ι.app j ≫
        ((J.plusCompIso F P).hom ≫ J.plusMap (whiskerLeft P η)).app X
state_after: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ (F.mapCocone (colimit.cocone (J.diagram P X.unop))).ι.app W ≫
      (whiskerLeft (J.plusObj P) η ≫ (J.plusCompIso G P).hom).app X =
    (F.mapCocone (colimit.cocone (J.diagram P X.unop))).ι.app W ≫
      ((J.plusCompIso F P).hom ≫ J.plusMap (whiskerLeft P η)).app X

tactic: dsimp [plusObj, plusMap]
state_before: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ (F.mapCocone (colimit.cocone (J.diagram P X.unop))).ι.app W ≫
      (whiskerLeft (J.plusObj P) η ≫ (J.plusCompIso G P).hom).app X =
    (F.mapCocone (colimit.cocone (J.diagram P X.unop))).ι.app W ≫
      ((J.plusCompIso F P).hom ≫ J.plusMap (whiskerLeft P η)).app X
state_after: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ F.map (colimit.ι (J.diagram P X.unop) W) ≫ η.app (colimit (J.diagram P X.unop)) ≫ (J.plusCompIso G P).hom.app X =
    F.map (colimit.ι (J.diagram P X.unop) W) ≫
      (J.plusCompIso F P).hom.app X ≫ colimMap (J.diagramNatTrans (whiskerLeft P η) X.unop)

tactic: simp only [ι_plusCompIso_hom, ι_colimMap, whiskerLeft_app, ι_plusCompIso_hom_assoc,
  NatTrans.naturality_assoc, GrothendieckTopology.diagramNatTrans_app]
state_before: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ F.map (colimit.ι (J.diagram P X.unop) W) ≫ η.app (colimit (J.diagram P X.unop)) ≫ (J.plusCompIso G P).hom.app X =
    F.map (colimit.ι (J.diagram P X.unop) W) ≫
      (J.plusCompIso F P).hom.app X ≫ colimMap (J.diagramNatTrans (whiskerLeft P η) X.unop)
state_after: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ η.app (multiequalizer (W.unop.index P)) ≫
      (J.diagramCompIso G P X.unop).hom.app W ≫ colimit.ι (J.diagram (P ⋙ G) X.unop) W =
    (J.diagramCompIso F P X.unop).hom.app W ≫
      Multiequalizer.lift (W.unop.index (P ⋙ G)) ((J.diagram (P ⋙ F) X.unop).obj W)
          (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯ ≫
        colimit.ι (J.diagram (P ⋙ G) X.unop) W

tactic: simp only [← Category.assoc]
state_before: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ η.app (multiequalizer (W.unop.index P)) ≫
      (J.diagramCompIso G P X.unop).hom.app W ≫ colimit.ι (J.diagram (P ⋙ G) X.unop) W =
    (J.diagramCompIso F P X.unop).hom.app W ≫
      Multiequalizer.lift (W.unop.index (P ⋙ G)) ((J.diagram (P ⋙ F) X.unop).obj W)
          (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯ ≫
        colimit.ι (J.diagram (P ⋙ G) X.unop) W
state_after: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ (η.app (multiequalizer (W.unop.index P)) ≫ (J.diagramCompIso G P X.unop).hom.app W) ≫
      colimit.ι (J.diagram (P ⋙ G) X.unop) W =
    ((J.diagramCompIso F P X.unop).hom.app W ≫
        Multiequalizer.lift (W.unop.index (P ⋙ G)) ((J.diagram (P ⋙ F) X.unop).obj W)
          (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯) ≫
      colimit.ι (J.diagram (P ⋙ G) X.unop) W

tactic: congr 1
state_before: case w.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ (η.app (multiequalizer (W.unop.index P)) ≫ (J.diagramCompIso G P X.unop).hom.app W) ≫
      colimit.ι (J.diagram (P ⋙ G) X.unop) W =
    ((J.diagramCompIso F P X.unop).hom.app W ≫
        Multiequalizer.lift (W.unop.index (P ⋙ G)) ((J.diagram (P ⋙ F) X.unop).obj W)
          (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯) ≫
      colimit.ι (J.diagram (P ⋙ G) X.unop) W
state_after: case w.h.e_a
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ η.app (multiequalizer (W.unop.index P)) ≫ (J.diagramCompIso G P X.unop).hom.app W =
    (J.diagramCompIso F P X.unop).hom.app W ≫
      Multiequalizer.lift (W.unop.index (P ⋙ G)) ((J.diagram (P ⋙ F) X.unop).obj W)
        (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯

tactic: apply Multiequalizer.hom_ext
state_before: case w.h.e_a
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ η.app (multiequalizer (W.unop.index P)) ≫ (J.diagramCompIso G P X.unop).hom.app W =
    (J.diagramCompIso F P X.unop).hom.app W ≫
      Multiequalizer.lift (W.unop.index (P ⋙ G)) ((J.diagram (P ⋙ F) X.unop).obj W)
        (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯
state_after: case w.h.e_a.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ ∀ (a : (W.unop.index (P ⋙ G)).L),
    (η.app (multiequalizer (W.unop.index P)) ≫ (J.diagramCompIso G P X.unop).hom.app W) ≫
        Multiequalizer.ι (W.unop.index (P ⋙ G)) a =
      ((J.diagramCompIso F P X.unop).hom.app W ≫
          Multiequalizer.lift (W.unop.index (P ⋙ G)) ((J.diagram (P ⋙ F) X.unop).obj W)
            (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯) ≫
        Multiequalizer.ι (W.unop.index (P ⋙ G)) a

tactic: intro a
state_before: case w.h.e_a.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
⊢ ∀ (a : (W.unop.index (P ⋙ G)).L),
    (η.app (multiequalizer (W.unop.index P)) ≫ (J.diagramCompIso G P X.unop).hom.app W) ≫
        Multiequalizer.ι (W.unop.index (P ⋙ G)) a =
      ((J.diagramCompIso F P X.unop).hom.app W ≫
          Multiequalizer.lift (W.unop.index (P ⋙ G)) ((J.diagram (P ⋙ F) X.unop).obj W)
            (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯) ≫
        Multiequalizer.ι (W.unop.index (P ⋙ G)) a
state_after: case w.h.e_a.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
a : (W.unop.index (P ⋙ G)).L
⊢ (η.app (multiequalizer (W.unop.index P)) ≫ (J.diagramCompIso G P X.unop).hom.app W) ≫
      Multiequalizer.ι (W.unop.index (P ⋙ G)) a =
    ((J.diagramCompIso F P X.unop).hom.app W ≫
        Multiequalizer.lift (W.unop.index (P ⋙ G)) ((J.diagram (P ⋙ F) X.unop).obj W)
          (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯) ≫
      Multiequalizer.ι (W.unop.index (P ⋙ G)) a

tactic: dsimp
state_before: case w.h.e_a.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
a : (W.unop.index (P ⋙ G)).L
⊢ (η.app (multiequalizer (W.unop.index P)) ≫ (J.diagramCompIso G P X.unop).hom.app W) ≫
      Multiequalizer.ι (W.unop.index (P ⋙ G)) a =
    ((J.diagramCompIso F P X.unop).hom.app W ≫
        Multiequalizer.lift (W.unop.index (P ⋙ G)) ((J.diagram (P ⋙ F) X.unop).obj W)
          (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯) ≫
      Multiequalizer.ι (W.unop.index (P ⋙ G)) a
state_after: case w.h.e_a.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
a : (W.unop.index (P ⋙ G)).L
⊢ (η.app (multiequalizer (W.unop.index P)) ≫ (J.diagramCompIso G P X.unop).hom.app W) ≫
      Multiequalizer.ι (W.unop.index (P ⋙ G)) a =
    ((J.diagramCompIso F P X.unop).hom.app W ≫
        Multiequalizer.lift (W.unop.index (P ⋙ G)) (multiequalizer (W.unop.index (P ⋙ F)))
          (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯) ≫
      Multiequalizer.ι (W.unop.index (P ⋙ G)) a

tactic: simp
state_before: case w.h.e_a.h
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝
P✝ : Cᵒᵖ ⥤ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝
F G : D ⥤ E
η : F ⟶ G
P : Cᵒᵖ ⥤ D
inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G
inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G
X : Cᵒᵖ
W : (J.Cover X.unop)ᵒᵖ
a : (W.unop.index (P ⋙ G)).L
⊢ (η.app (multiequalizer (W.unop.index P)) ≫ (J.diagramCompIso G P X.unop).hom.app W) ≫
      Multiequalizer.ι (W.unop.index (P ⋙ G)) a =
    ((J.diagramCompIso F P X.unop).hom.app W ≫
        Multiequalizer.lift (W.unop.index (P ⋙ G)) (multiequalizer (W.unop.index (P ⋙ F)))
          (fun i => Multiequalizer.ι (W.unop.index (P ⋙ F)) i ≫ η.app (P.obj { unop := i.Y })) ⋯) ≫
      Multiequalizer.ι (W.unop.index (P ⋙ G)) a
state_after: no goals