Theorem: CategoryTheory.GrothendieckTopology.coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation_π_app
File: Mathlib/CategoryTheory/Sites/LeftExact.lean

Proof steps:
state_before: 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 : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type (max v u)
inst✝ : SmallCategory K
F : K ⥤ Cᵒᵖ ⥤ D
W : J.Cover X
i : W.Arrow
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W))
⊢ ∀ ⦃X_1 Y : K⦄ (f : X_1 ⟶ Y),
    ((Functor.const K).obj E.pt).map f ≫ (fun k => E.π.app k ≫ Multiequalizer.ι (W.index (F.obj k)) i) Y =
      (fun k => E.π.app k ≫ Multiequalizer.ι (W.index (F.obj k)) i) X_1 ≫ (F ⋙ (evaluation Cᵒᵖ D).obj (op i.Y)).map f
state_after: 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 : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type (max v u)
inst✝ : SmallCategory K
F : K ⥤ Cᵒᵖ ⥤ D
W : J.Cover X
i : W.Arrow
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W))
a b : K
f : a ⟶ b
⊢ ((Functor.const K).obj E.pt).map f ≫ (fun k => E.π.app k ≫ Multiequalizer.ι (W.index (F.obj k)) i) b =
    (fun k => E.π.app k ≫ Multiequalizer.ι (W.index (F.obj k)) i) a ≫ (F ⋙ (evaluation Cᵒᵖ D).obj (op i.Y)).map f
tactic: intro a b f
state_before: 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 : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type (max v u)
inst✝ : SmallCategory K
F : K ⥤ Cᵒᵖ ⥤ D
W : J.Cover X
i : W.Arrow
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W))
a b : K
f : a ⟶ b
⊢ ((Functor.const K).obj E.pt).map f ≫ (fun k => E.π.app k ≫ Multiequalizer.ι (W.index (F.obj k)) i) b =
    (fun k => E.π.app k ≫ Multiequalizer.ι (W.index (F.obj k)) i) a ≫ (F ⋙ (evaluation Cᵒᵖ D).obj (op i.Y)).map f
state_after: 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 : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type (max v u)
inst✝ : SmallCategory K
F : K ⥤ Cᵒᵖ ⥤ D
W : J.Cover X
i : W.Arrow
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W))
a b : K
f : a ⟶ b
⊢ 𝟙 E.pt ≫ E.π.app b ≫ Multiequalizer.ι (W.index (F.obj b)) i =
    (E.π.app a ≫ Multiequalizer.ι (W.index (F.obj a)) i) ≫ (F.map f).app (op i.Y)
tactic: dsimp
state_before: 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 : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type (max v u)
inst✝ : SmallCategory K
F : K ⥤ Cᵒᵖ ⥤ D
W : J.Cover X
i : W.Arrow
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W))
a b : K
f : a ⟶ b
⊢ 𝟙 E.pt ≫ E.π.app b ≫ Multiequalizer.ι (W.index (F.obj b)) i =
    (E.π.app a ≫ Multiequalizer.ι (W.index (F.obj a)) i) ≫ (F.map f).app (op i.Y)
state_after: 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 : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type (max v u)
inst✝ : SmallCategory K
F : K ⥤ Cᵒᵖ ⥤ D
W : J.Cover X
i : W.Arrow
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W))
a b : K
f : a ⟶ b
⊢ (E.π.app a ≫ (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W)).map f) ≫
      Multiequalizer.ι (W.index (F.obj b)) i =
    E.π.app a ≫ Multiequalizer.ι (W.index (F.obj a)) i ≫ (F.map f).app (op i.Y)
tactic: rw [Category.id_comp, Category.assoc, ← E.w f]
state_before: 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 : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type (max v u)
inst✝ : SmallCategory K
F : K ⥤ Cᵒᵖ ⥤ D
W : J.Cover X
i : W.Arrow
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W))
a b : K
f : a ⟶ b
⊢ (E.π.app a ≫ (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W)).map f) ≫
      Multiequalizer.ι (W.index (F.obj b)) i =
    E.π.app a ≫ Multiequalizer.ι (W.index (F.obj a)) i ≫ (F.map f).app (op i.Y)
state_after: 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 : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type (max v u)
inst✝ : SmallCategory K
F : K ⥤ Cᵒᵖ ⥤ D
W : J.Cover X
i : W.Arrow
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W))
a b : K
f : a ⟶ b
⊢ (E.π.app a ≫
        Multiequalizer.lift (W.index (F.obj b)) (multiequalizer (W.index (F.obj a)))
          (fun i => Multiequalizer.ι (W.index (F.obj a)) i ≫ (F.map f).app (op i.Y)) ⋯) ≫
      Multiequalizer.ι (W.index (F.obj b)) i =
    E.π.app a ≫ Multiequalizer.ι (W.index (F.obj a)) i ≫ (F.map f).app (op i.Y)
tactic: dsimp [diagramNatTrans]
state_before: 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 : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type (max v u)
inst✝ : SmallCategory K
F : K ⥤ Cᵒᵖ ⥤ D
W : J.Cover X
i : W.Arrow
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj (op W))
a b : K
f : a ⟶ b
⊢ (E.π.app a ≫
        Multiequalizer.lift (W.index (F.obj b)) (multiequalizer (W.index (F.obj a)))
          (fun i => Multiequalizer.ι (W.index (F.obj a)) i ≫ (F.map f).app (op i.Y)) ⋯) ≫
      Multiequalizer.ι (W.index (F.obj b)) i =
    E.π.app a ≫ Multiequalizer.ι (W.index (F.obj a)) i ≫ (F.map f).app (op i.Y)
state_after: no goals
tactic: simp only [Multiequalizer.lift_ι, Category.assoc]