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

tactic: ext X
state_before: C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
⊢ whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom = (plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)

tactic: apply (isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).hom_ext
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
⊢ (whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom).app X =
    ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)).app X

tactic: intro W
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
⊢ ∀ (j : (Cover J X.unop)ᵒᵖ),
    (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app j ≫
        (whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom).app X =
      (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app j ≫
        ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)).app X

tactic: dsimp [plusObj, plusMap]
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app W ≫
      (whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom).app X =
    (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app W ≫
      ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)).app X

tactic: simp only [ι_colimMap, whiskerRight_app, ι_plusCompIso_hom_assoc,
  GrothendieckTopology.diagramNatTrans_app]
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ F.map (colimit.ι (diagram J P X.unop) W) ≫
      F.map (colimMap (diagramNatTrans J η X.unop)) ≫ (plusCompIso J F Q).hom.app X =
    F.map (colimit.ι (diagram J P X.unop) W) ≫
      (plusCompIso J F P).hom.app X ≫ colimMap (diagramNatTrans J (whiskerRight η F) X.unop)

tactic: simp only [← Category.assoc, ← F.map_comp]
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ F.map (colimit.ι (diagram J P X.unop) W) ≫
      F.map (colimMap (diagramNatTrans J η X.unop)) ≫ (plusCompIso J F Q).hom.app X =
    (diagramCompIso J F P X.unop).hom.app W ≫
      Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) ((diagram J (P ⋙ F) X.unop).obj W)
          (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
          (_ :
            ∀ (b : (Cover.index W.unop (Q ⋙ F)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
                    (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) b) ≫
                  MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) b =
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
                    (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) b) ≫
                  MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) b) ≫
        colimit.ι (diagram J (Q ⋙ F) X.unop) W

tactic: dsimp [colimMap, IsColimit.map]
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ F.map (colimit.ι (diagram J P X.unop) W ≫ colimMap (diagramNatTrans J η X.unop)) ≫ (plusCompIso J F Q).hom.app X =
    ((diagramCompIso J F P X.unop).hom.app W ≫
        Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) ((diagram J (P ⋙ F) X.unop).obj W)
          (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
          (_ :
            ∀ (b : (Cover.index W.unop (Q ⋙ F)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
                    (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) b) ≫
                  MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) b =
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
                    (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) b) ≫
                  MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) b)) ≫
      colimit.ι (diagram J (Q ⋙ F) X.unop) W

tactic: simp only [colimit.ι_desc]
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ F.map
        (colimit.ι (diagram J P X.unop) W ≫
          colimit.desc (diagram J P X.unop)
            ((Cocones.precompose (diagramNatTrans J η X.unop)).obj (colimit.cocone (diagram J Q X.unop)))) ≫
      (plusCompIso J F Q).hom.app X =
    ((diagramCompIso J F P X.unop).hom.app W ≫
        Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))
          (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
          (_ :
            ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫
      colimit.ι (diagram J (Q ⋙ F) X.unop) W

tactic: dsimp [Cocones.precompose]
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ F.map (((Cocones.precompose (diagramNatTrans J η X.unop)).obj (colimit.cocone (diagram J Q X.unop))).ι.app W) ≫
      (plusCompIso J F Q).hom.app X =
    ((diagramCompIso J F P X.unop).hom.app W ≫
        Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))
          (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
          (_ :
            ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫
      colimit.ι (diagram J (Q ⋙ F) X.unop) W

tactic: simp only [Functor.map_comp, Category.assoc, ι_plusCompIso_hom]
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ F.map
        (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))
            (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
            (_ :
              ∀ (i : (Cover.index W.unop Q).R),
                (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                      (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫
                    MulticospanIndex.fst (Cover.index W.unop Q) i =
                  (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                      (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫
                    MulticospanIndex.snd (Cover.index W.unop Q) i) ≫
          colimit.ι (diagram J Q X.unop) W) ≫
      (plusCompIso J F Q).hom.app X =
    ((diagramCompIso J F P X.unop).hom.app W ≫
        Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))
          (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
          (_ :
            ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫
      colimit.ι (diagram J (Q ⋙ F) X.unop) W

tactic: simp only [← Category.assoc]
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ F.map
        (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))
          (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
          (_ :
            ∀ (i : (Cover.index W.unop Q).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                    (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop Q) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                    (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫
      (diagramCompIso J F Q X.unop).hom.app W ≫ colimit.ι (diagram J (Q ⋙ F) X.unop) W =
    (diagramCompIso J F P X.unop).hom.app W ≫
      Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))
          (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
          (_ :
            ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i) ≫
        colimit.ι (diagram J (Q ⋙ F) X.unop) W

tactic: congr 1
state_before: case w.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ (F.map
          (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))
            (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
            (_ :
              ∀ (i : (Cover.index W.unop Q).R),
                (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                      (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫
                    MulticospanIndex.fst (Cover.index W.unop Q) i =
                  (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                      (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫
                    MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫
        (diagramCompIso J F Q X.unop).hom.app W) ≫
      colimit.ι (diagram J (Q ⋙ F) X.unop) W =
    ((diagramCompIso J F P X.unop).hom.app W ≫
        Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))
          (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
          (_ :
            ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫
      colimit.ι (diagram J (Q ⋙ F) X.unop) W

tactic: apply Multiequalizer.hom_ext
state_before: case w.h.e_a
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ F.map
        (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))
          (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
          (_ :
            ∀ (i : (Cover.index W.unop Q).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                    (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop Q) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                    (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫
      (diagramCompIso J F Q X.unop).hom.app W =
    (diagramCompIso J F P X.unop).hom.app W ≫
      Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))
        (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
        (_ :
          ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),
            (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                  (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                  (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)

tactic: intro a
state_before: case w.h.e_a.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
⊢ ∀ (a : (Cover.index W.unop (Q ⋙ F)).L),
    (F.map
            (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))
              (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
              (_ :
                ∀ (i : (Cover.index W.unop Q).R),
                  (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                        (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫
                      MulticospanIndex.fst (Cover.index W.unop Q) i =
                    (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                        (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫
                      MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫
          (diagramCompIso J F Q X.unop).hom.app W) ≫
        Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a =
      ((diagramCompIso J F P X.unop).hom.app W ≫
          Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))
            (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
            (_ :
              ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                      (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                    MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =
                  (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                      (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                    MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫
        Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a

tactic: dsimp
state_before: case w.h.e_a.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
a : (Cover.index W.unop (Q ⋙ F)).L
⊢ (F.map
          (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))
            (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
            (_ :
              ∀ (i : (Cover.index W.unop Q).R),
                (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                      (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫
                    MulticospanIndex.fst (Cover.index W.unop Q) i =
                  (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                      (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫
                    MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫
        (diagramCompIso J F Q X.unop).hom.app W) ≫
      Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a =
    ((diagramCompIso J F P X.unop).hom.app W ≫
        Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))
          (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
          (_ :
            ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫
      Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a

tactic: simp only [diagramCompIso_hom_ι_assoc, Multiequalizer.lift_ι, diagramCompIso_hom_ι,
  Category.assoc]
state_before: case w.h.e_a.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
a : (Cover.index W.unop (Q ⋙ F)).L
⊢ (F.map
          (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))
            (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
            (_ :
              ∀ (i : (Cover.index W.unop Q).R),
                (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                      (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫
                    MulticospanIndex.fst (Cover.index W.unop Q) i =
                  (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                      (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫
                    MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫
        (diagramCompIso J F Q X.unop).hom.app W) ≫
      Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a =
    ((diagramCompIso J F P X.unop).hom.app W ≫
        Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))
          (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))
          (_ :
            ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)
                    (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫
      Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a

tactic: simp only [← F.map_comp, Multiequalizer.lift_ι]
state_before: case w.h.e_a.h
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category 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
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
a : (Cover.index W.unop (Q ⋙ F)).L
⊢ F.map
        (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))
          (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
          (_ :
            ∀ (i : (Cover.index W.unop Q).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                    (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop Q) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)
                    (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫
      F.map (Multiequalizer.ι (Cover.index W.unop Q) a) =
    F.map (Multiequalizer.ι (Cover.index W.unop P) a) ≫ F.map (η.app a.Y.op)