Theorem: CategoryTheory.GrothendieckTopology.diagramNatTrans_comp
File: Mathlib/CategoryTheory/Sites/Plus.lean

Proof steps:
state_before: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹ : Category.{max v u, w} D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P✝ P Q R : Cᵒᵖ ⥤ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
⊢ J.diagramNatTrans (η ≫ γ) X = J.diagramNatTrans η X ≫ J.diagramNatTrans γ X
state_after: case w.h
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹ : Category.{max v u, w} D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P✝ P Q R : Cᵒᵖ ⥤ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
x✝ : (J.Cover X)ᵒᵖ
⊢ (J.diagramNatTrans (η ≫ γ) X).app x✝ = (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X).app x✝
tactic: ext : 2
state_before: case w.h
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹ : Category.{max v u, w} D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P✝ P Q R : Cᵒᵖ ⥤ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
x✝ : (J.Cover X)ᵒᵖ
⊢ (J.diagramNatTrans (η ≫ γ) X).app x✝ = (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X).app x✝
state_after: case w.h
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹ : Category.{max v u, w} D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P✝ P Q R : Cᵒᵖ ⥤ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
x✝ : (J.Cover X)ᵒᵖ
i : ((unop x✝).index R).L
⊢ (J.diagramNatTrans (η ≫ γ) X).app x✝ ≫ Multiequalizer.ι ((unop x✝).index R) i =
    (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X).app x✝ ≫ Multiequalizer.ι ((unop x✝).index R) i
tactic: refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
state_before: case w.h
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹ : Category.{max v u, w} D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P✝ P Q R : Cᵒᵖ ⥤ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
x✝ : (J.Cover X)ᵒᵖ
i : ((unop x✝).index R).L
⊢ (J.diagramNatTrans (η ≫ γ) X).app x✝ ≫ Multiequalizer.ι ((unop x✝).index R) i =
    (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X).app x✝ ≫ Multiequalizer.ι ((unop x✝).index R) i
state_after: case w.h
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹ : Category.{max v u, w} D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P✝ P Q R : Cᵒᵖ ⥤ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
x✝ : (J.Cover X)ᵒᵖ
i : ((unop x✝).index R).L
⊢ Multiequalizer.lift ((unop x✝).index R) (multiequalizer ((unop x✝).index P))
        (fun i => Multiequalizer.ι ((unop x✝).index P) i ≫ η.app (op i.Y) ≫ γ.app (op i.Y)) ⋯ ≫
      Multiequalizer.ι ((unop x✝).index R) i =
    (Multiequalizer.lift ((unop x✝).index Q) (multiequalizer ((unop x✝).index P))
          (fun i => Multiequalizer.ι ((unop x✝).index P) i ≫ η.app (op i.Y)) ⋯ ≫
        Multiequalizer.lift ((unop x✝).index R) (multiequalizer ((unop x✝).index Q))
          (fun i => Multiequalizer.ι ((unop x✝).index Q) i ≫ γ.app (op i.Y)) ⋯) ≫
      Multiequalizer.ι ((unop x✝).index R) i
tactic: dsimp
state_before: case w.h
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹ : Category.{max v u, w} D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P✝ P Q R : Cᵒᵖ ⥤ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
x✝ : (J.Cover X)ᵒᵖ
i : ((unop x✝).index R).L
⊢ Multiequalizer.lift ((unop x✝).index R) (multiequalizer ((unop x✝).index P))
        (fun i => Multiequalizer.ι ((unop x✝).index P) i ≫ η.app (op i.Y) ≫ γ.app (op i.Y)) ⋯ ≫
      Multiequalizer.ι ((unop x✝).index R) i =
    (Multiequalizer.lift ((unop x✝).index Q) (multiequalizer ((unop x✝).index P))
          (fun i => Multiequalizer.ι ((unop x✝).index P) i ≫ η.app (op i.Y)) ⋯ ≫
        Multiequalizer.lift ((unop x✝).index R) (multiequalizer ((unop x✝).index Q))
          (fun i => Multiequalizer.ι ((unop x✝).index Q) i ≫ γ.app (op i.Y)) ⋯) ≫
      Multiequalizer.ι ((unop x✝).index R) i
state_after: no goals
tactic: simp