l3lab/ntp-mathlib huggingface dataset:
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  

Proof state:
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
⊢ GrothendieckTopology.toSheafify J (G.op ⋙ F) ≫
      (coyoneda.preimage
            ((Adjunction.leftAdjointsCoyonedaEquiv
                    (Adjunction.comp (sheafificationAdjunction K A) (sheafAdjunctionCocontinuous G A J K))
                    (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A))).hom.app
              (op F))).unop.val =
    whiskerLeft G.op (GrothendieckTopology.toSheafify K F)

Next tactic:
apply Quiver.Hom.op_inj
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  

Proof state:
case a
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
⊢ (GrothendieckTopology.toSheafify J (G.op ⋙ F) ≫
        (coyoneda.preimage
              ((Adjunction.leftAdjointsCoyonedaEquiv
                      (Adjunction.comp (sheafificationAdjunction K A) (sheafAdjunctionCocontinuous G A J K))
                      (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A))).hom.app
                (op F))).unop.val).op =
    (whiskerLeft G.op (GrothendieckTopology.toSheafify K F)).op

Next tactic:
apply coyoneda.map_injective
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  

Proof state:
case a.a
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
⊢ coyoneda.map
      (GrothendieckTopology.toSheafify J (G.op ⋙ F) ≫
          (coyoneda.preimage
                ((Adjunction.leftAdjointsCoyonedaEquiv
                        (Adjunction.comp (sheafificationAdjunction K A) (sheafAdjunctionCocontinuous G A J K))
                        (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A))).hom.app
                  (op F))).unop.val).op =
    coyoneda.map (whiskerLeft G.op (GrothendieckTopology.toSheafify K F)).op

Next tactic:
ext E : 2
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  ext E : 2
  

Proof state:
case a.a.w.h
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
E : Cᵒᵖ ⥤ A
⊢ (coyoneda.map
          (GrothendieckTopology.toSheafify J (G.op ⋙ F) ≫
              (coyoneda.preimage
                    ((Adjunction.leftAdjointsCoyonedaEquiv
                            (Adjunction.comp (sheafificationAdjunction K A) (sheafAdjunctionCocontinuous G A J K))
                            (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A))).hom.app
                      (op F))).unop.val).op).app
      E =
    (coyoneda.map (whiskerLeft G.op (GrothendieckTopology.toSheafify K F)).op).app E

Next tactic:
dsimp [Functor.preimage, Full.preimage, coyoneda, Adjunction.leftAdjointsCoyonedaEquiv]
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  ext E : 2
  dsimp [Functor.preimage, Full.preimage, coyoneda, Adjunction.leftAdjointsCoyonedaEquiv]
  

Proof state:
case a.a.w.h
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
E : Cᵒᵖ ⥤ A
⊢ (fun g =>
      (GrothendieckTopology.toSheafify J (G.op ⋙ F) ≫
          ((Adjunction.homEquiv (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A)) F
                  ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F))).symm
              ((Adjunction.homEquiv
                  (Adjunction.comp (sheafificationAdjunction K A) (sheafAdjunctionCocontinuous G A J K)) F
                  ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)))
                (𝟙 ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F))))).val) ≫
        g) =
    fun g => whiskerLeft G.op (GrothendieckTopology.toSheafify K F) ≫ g

Next tactic:
erw [Adjunction.homEquiv_unit, Adjunction.homEquiv_counit]
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  ext E : 2
  dsimp [Functor.preimage, Full.preimage, coyoneda, Adjunction.leftAdjointsCoyonedaEquiv]
  erw [Adjunction.homEquiv_unit, Adjunction.homEquiv_counit]
  

Proof state:
case a.a.w.h
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
E : Cᵒᵖ ⥤ A
⊢ (fun g =>
      (GrothendieckTopology.toSheafify J (G.op ⋙ F) ≫
          (((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op ⋙ presheafToSheaf J A).map
                ((Adjunction.comp (sheafificationAdjunction K A) (sheafAdjunctionCocontinuous G A J K)).unit.app F ≫
                  (sheafToPresheaf J A ⋙ ran G.op).map
                    (𝟙 ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)))) ≫
              (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A)).counit.app
                ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F))).val) ≫
        g) =
    fun g => whiskerLeft G.op (GrothendieckTopology.toSheafify K F) ≫ g

Next tactic:
dsimp [Adjunction.comp]
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  ext E : 2
  dsimp [Functor.preimage, Full.preimage, coyoneda, Adjunction.leftAdjointsCoyonedaEquiv]
  erw [Adjunction.homEquiv_unit, Adjunction.homEquiv_counit]
  dsimp [Adjunction.comp]
  

Proof state:
case a.a.w.h
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
E : Cᵒᵖ ⥤ A
⊢ (fun g =>
      (GrothendieckTopology.toSheafify J (G.op ⋙ F) ≫
          GrothendieckTopology.sheafifyMap J
              ((whiskerLeft G.op ((sheafificationAdjunction K A).unit.app F) ≫
                  whiskerLeft G.op ((Ran.adjunction A G.op).unit.app (GrothendieckTopology.sheafify K F)) ≫
                    𝟙
                      (G.op ⋙
                        ((sheafPushforwardCocontinuous G A J K).obj
                            ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F))).val)) ≫
                whiskerLeft G.op
                  ((ran G.op).map (𝟙 ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val))) ≫
            𝟙
                (GrothendieckTopology.sheafify J
                  (G.op ⋙
                    (ran G.op).obj ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val)) ≫
              GrothendieckTopology.sheafifyMap J
                  ((Ran.adjunction A G.op).counit.app
                    ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val) ≫
                GrothendieckTopology.sheafifyLift J
                  (𝟙 ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val)
                  (_ :
                    Presheaf.IsSheaf J ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val)) ≫
        g) =
    fun g => whiskerLeft G.op (GrothendieckTopology.toSheafify K F) ≫ g

Next tactic:
simp only [sheafificationAdjunction_unit_app, Category.comp_id, Functor.map_id,
    whiskerLeft_id', GrothendieckTopology.sheafifyMap_comp,
    GrothendieckTopology.sheafifyMap_sheafifyLift, Category.id_comp,
    Category.assoc, GrothendieckTopology.toSheafify_sheafifyLift]
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  ext E : 2
  dsimp [Functor.preimage, Full.preimage, coyoneda, Adjunction.leftAdjointsCoyonedaEquiv]
  erw [Adjunction.homEquiv_unit, Adjunction.homEquiv_counit]
  dsimp [Adjunction.comp]
  simp only [sheafificationAdjunction_unit_app, Category.comp_id, Functor.map_id,
    whiskerLeft_id', GrothendieckTopology.sheafifyMap_comp,
    GrothendieckTopology.sheafifyMap_sheafifyLift, Category.id_comp,
    Category.assoc, GrothendieckTopology.toSheafify_sheafifyLift]
  

Proof state:
case a.a.w.h
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
E : Cᵒᵖ ⥤ A
⊢ (fun g =>
      whiskerLeft G.op (GrothendieckTopology.toSheafify K F) ≫
        whiskerLeft G.op ((Ran.adjunction A G.op).unit.app (GrothendieckTopology.sheafify K F)) ≫
          (Ran.adjunction A G.op).counit.app
              ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val ≫
            g) =
    fun g => whiskerLeft G.op (GrothendieckTopology.toSheafify K F) ≫ g

Next tactic:
ext t s : 3
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  ext E : 2
  dsimp [Functor.preimage, Full.preimage, coyoneda, Adjunction.leftAdjointsCoyonedaEquiv]
  erw [Adjunction.homEquiv_unit, Adjunction.homEquiv_counit]
  dsimp [Adjunction.comp]
  simp only [sheafificationAdjunction_unit_app, Category.comp_id, Functor.map_id,
    whiskerLeft_id', GrothendieckTopology.sheafifyMap_comp,
    GrothendieckTopology.sheafifyMap_sheafifyLift, Category.id_comp,
    Category.assoc, GrothendieckTopology.toSheafify_sheafifyLift]
  ext t s : 3
  

Proof state:
case a.a.w.h.h.w.h
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
E : Cᵒᵖ ⥤ A
t : ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val ⟶ E
s : Cᵒᵖ
⊢ (whiskerLeft G.op (GrothendieckTopology.toSheafify K F) ≫
          whiskerLeft G.op ((Ran.adjunction A G.op).unit.app (GrothendieckTopology.sheafify K F)) ≫
            (Ran.adjunction A G.op).counit.app
                ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val ≫
              t).app
      s =
    (whiskerLeft G.op (GrothendieckTopology.toSheafify K F) ≫ t).app s

Next tactic:
dsimp [sheafPushforwardContinuous]
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  ext E : 2
  dsimp [Functor.preimage, Full.preimage, coyoneda, Adjunction.leftAdjointsCoyonedaEquiv]
  erw [Adjunction.homEquiv_unit, Adjunction.homEquiv_counit]
  dsimp [Adjunction.comp]
  simp only [sheafificationAdjunction_unit_app, Category.comp_id, Functor.map_id,
    whiskerLeft_id', GrothendieckTopology.sheafifyMap_comp,
    GrothendieckTopology.sheafifyMap_sheafifyLift, Category.id_comp,
    Category.assoc, GrothendieckTopology.toSheafify_sheafifyLift]
  ext t s : 3
  dsimp [sheafPushforwardContinuous]
  

Proof state:
case a.a.w.h.h.w.h
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
E : Cᵒᵖ ⥤ A
t : ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val ⟶ E
s : Cᵒᵖ
⊢ (GrothendieckTopology.toSheafify K F).app (op (G.obj s.unop)) ≫
      ((Ran.adjunction A G.op).unit.app (GrothendieckTopology.sheafify K F)).app (op (G.obj s.unop)) ≫
        ((Ran.adjunction A G.op).counit.app (G.op ⋙ GrothendieckTopology.sheafify K F)).app s ≫ t.app s =
    (GrothendieckTopology.toSheafify K F).app (op (G.obj s.unop)) ≫ t.app s

Next tactic:
congr 1
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  ext E : 2
  dsimp [Functor.preimage, Full.preimage, coyoneda, Adjunction.leftAdjointsCoyonedaEquiv]
  erw [Adjunction.homEquiv_unit, Adjunction.homEquiv_counit]
  dsimp [Adjunction.comp]
  simp only [sheafificationAdjunction_unit_app, Category.comp_id, Functor.map_id,
    whiskerLeft_id', GrothendieckTopology.sheafifyMap_comp,
    GrothendieckTopology.sheafifyMap_sheafifyLift, Category.id_comp,
    Category.assoc, GrothendieckTopology.toSheafify_sheafifyLift]
  ext t s : 3
  dsimp [sheafPushforwardContinuous]
  congr 1
  

Proof state:
case a.a.w.h.h.w.h.e_a
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
E : Cᵒᵖ ⥤ A
t : ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val ⟶ E
s : Cᵒᵖ
⊢ ((Ran.adjunction A G.op).unit.app (GrothendieckTopology.sheafify K F)).app (op (G.obj s.unop)) ≫
      ((Ran.adjunction A G.op).counit.app (G.op ⋙ GrothendieckTopology.sheafify K F)).app s ≫ t.app s =
    t.app s

Next tactic:
simp only [← Category.assoc]
lemma Functor.toSheafify_pullbackSheafificationCompatibility (F : Dᵒᵖ ⥤ A) :
    J.toSheafify (G.op ⋙ F) ≫
    ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =
    whiskerLeft _ (K.toSheafify _) := by
  dsimp [pushforwardContinuousSheafificationCompatibility, Adjunction.leftAdjointUniq]
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  ext E : 2
  dsimp [Functor.preimage, Full.preimage, coyoneda, Adjunction.leftAdjointsCoyonedaEquiv]
  erw [Adjunction.homEquiv_unit, Adjunction.homEquiv_counit]
  dsimp [Adjunction.comp]
  simp only [sheafificationAdjunction_unit_app, Category.comp_id, Functor.map_id,
    whiskerLeft_id', GrothendieckTopology.sheafifyMap_comp,
    GrothendieckTopology.sheafifyMap_sheafifyLift, Category.id_comp,
    Category.assoc, GrothendieckTopology.toSheafify_sheafifyLift]
  ext t s : 3
  dsimp [sheafPushforwardContinuous]
  congr 1
  simp only [← Category.assoc]
  

Proof state:
case a.a.w.h.h.w.h.e_a
C D : Type u
inst✝¹³ : Category.{v, u} C
inst✝¹² : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝¹¹ : Category.{max u v, w} A
inst✝¹⁰ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝⁹ : IsCocontinuous G J K
inst✝⁸ : ConcreteCategory A
inst✝⁷ : PreservesLimits (forget A)
inst✝⁶ : ReflectsIsomorphisms (forget A)
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget A)
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ A
inst✝³ : (X : D) → PreservesColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ (forget A)
inst✝² : ∀ (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)ᵒᵖ A
inst✝¹ : IsCocontinuous G J K
inst✝ : IsContinuous G J K
F : Dᵒᵖ ⥤ A
E : Cᵒᵖ ⥤ A
t : ((sheafPushforwardContinuous G A J K).obj ((presheafToSheaf K A).obj F)).val ⟶ E
s : Cᵒᵖ
⊢ (((Ran.adjunction A G.op).unit.app (GrothendieckTopology.sheafify K F)).app (op (G.obj s.unop)) ≫
        ((Ran.adjunction A G.op).counit.app (G.op ⋙ GrothendieckTopology.sheafify K F)).app s) ≫
      t.app s =
    t.app s

Next tactic:
convert Category.id_comp (obj := A) _