Theorem: CategoryTheory.sheafComposeNatTrans_app_uniq
File: Mathlib/CategoryTheory/Sites/PreservesSheafification.lean

tactic: apply (adj₂.homEquiv _ _).injective
state_before: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝² : Category.{u_3, u_1} A
inst✝¹ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝ : J.HasSheafCompose F
P : Cᵒᵖ ⥤ A
α : G₂.obj (P ⋙ F) ⟶ (sheafCompose J F).obj (G₁.obj P)
hα : adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α = whiskerRight (adj₁.unit.app P) F
⊢ α = (sheafComposeNatTrans J F adj₁ adj₂).app P
state_after: case a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝² : Category.{u_3, u_1} A
inst✝¹ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝ : J.HasSheafCompose F
P : Cᵒᵖ ⥤ A
α : G₂.obj (P ⋙ F) ⟶ (sheafCompose J F).obj (G₁.obj P)
hα : adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α = whiskerRight (adj₁.unit.app P) F
⊢ (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))) α =
    (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))) ((sheafComposeNatTrans J F adj₁ adj₂).app P)

tactic: dsimp [sheafComposeNatTrans]
state_before: case a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝² : Category.{u_3, u_1} A
inst✝¹ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝ : J.HasSheafCompose F
P : Cᵒᵖ ⥤ A
α : G₂.obj (P ⋙ F) ⟶ (sheafCompose J F).obj (G₁.obj P)
hα : adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α = whiskerRight (adj₁.unit.app P) F
⊢ (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))) α =
    (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))) ((sheafComposeNatTrans J F adj₁ adj₂).app P)
state_after: case a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝² : Category.{u_3, u_1} A
inst✝¹ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝ : J.HasSheafCompose F
P : Cᵒᵖ ⥤ A
α : G₂.obj (P ⋙ F) ⟶ (sheafCompose J F).obj (G₁.obj P)
hα : adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α = whiskerRight (adj₁.unit.app P) F
⊢ (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))) α =
    (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P)))
      ((adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))).symm (whiskerRight (adj₁.unit.app P) F))

tactic: erw [Equiv.apply_symm_apply]
state_before: case a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝² : Category.{u_3, u_1} A
inst✝¹ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝ : J.HasSheafCompose F
P : Cᵒᵖ ⥤ A
α : G₂.obj (P ⋙ F) ⟶ (sheafCompose J F).obj (G₁.obj P)
hα : adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α = whiskerRight (adj₁.unit.app P) F
⊢ (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))) α =
    (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P)))
      ((adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))).symm (whiskerRight (adj₁.unit.app P) F))
state_after: case a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝² : Category.{u_3, u_1} A
inst✝¹ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝ : J.HasSheafCompose F
P : Cᵒᵖ ⥤ A
α : G₂.obj (P ⋙ F) ⟶ (sheafCompose J F).obj (G₁.obj P)
hα : adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α = whiskerRight (adj₁.unit.app P) F
⊢ (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))) α = whiskerRight (adj₁.unit.app P) F

tactic: rw [← hα]
state_before: case a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝² : Category.{u_3, u_1} A
inst✝¹ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝ : J.HasSheafCompose F
P : Cᵒᵖ ⥤ A
α : G₂.obj (P ⋙ F) ⟶ (sheafCompose J F).obj (G₁.obj P)
hα : adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α = whiskerRight (adj₁.unit.app P) F
⊢ (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))) α = whiskerRight (adj₁.unit.app P) F
state_after: case a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝² : Category.{u_3, u_1} A
inst✝¹ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝ : J.HasSheafCompose F
P : Cᵒᵖ ⥤ A
α : G₂.obj (P ⋙ F) ⟶ (sheafCompose J F).obj (G₁.obj P)
hα : adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α = whiskerRight (adj₁.unit.app P) F
⊢ (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))) α = adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α

tactic: apply adj₂.homEquiv_unit
state_before: case a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝² : Category.{u_3, u_1} A
inst✝¹ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝ : J.HasSheafCompose F
P : Cᵒᵖ ⥤ A
α : G₂.obj (P ⋙ F) ⟶ (sheafCompose J F).obj (G₁.obj P)
hα : adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α = whiskerRight (adj₁.unit.app P) F
⊢ (adj₂.homEquiv (P ⋙ F) ((sheafCompose J F).obj (G₁.obj P))) α = adj₂.unit.app (P ⋙ F) ≫ (sheafToPresheaf J B).map α
state_after: no goals