-- Mathlib/CategoryTheory/Sites/CoverPreserving.lean
theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
    (x.functorPushforward G).Compatible

Proof state:
case mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : Sheaf K (Type w)
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : x.Compatible
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
X₁ : C
f₁ : X₁ ⟶ Z
h₁ : Z₁ ⟶ G.obj X₁
hf₁ : T f₁
H₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)
X₂ : C
f₂ : X₂ ⟶ Z
h₂ : Z₂ ⟶ G.obj X₂
hf₂ : T f₂
H₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)
eq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂
⊢ ℱ.val.map g₁.op
      (FunctorPushforwardStructure.casesOn { preobj := X₁, premap := f₁, lift := h₁, cover := hf₁, fac := ⋯ }
        fun Z_1 g h h₁_1 fac => ℱ.val.map h.op (x g h₁_1)) =
    ℱ.val.map g₂.op
      (FunctorPushforwardStructure.casesOn { preobj := X₂, premap := f₂, lift := h₂, cover := hf₂, fac := ⋯ }
        fun Z_1 g h h₁ fac => ℱ.val.map h.op (x g h₁))

Tactic:
suffices ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂) by
  simpa using this

Result:
case mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : Sheaf K (Type w)
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : x.Compatible
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
X₁ : C
f₁ : X₁ ⟶ Z
h₁ : Z₁ ⟶ G.obj X₁
hf₁ : T f₁
H₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)
X₂ : C
f₂ : X₂ ⟶ Z
h₂ : Z₂ ⟶ G.obj X₂
hf₂ : T f₂
H₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)
eq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂
⊢ ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂)

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : Sheaf K (Type w)
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : x.Compatible
⊢ (FamilyOfElements.functorPushforward G x).Compatible

Tactic:
rintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq

Result:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : Sheaf K (Type w)
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : x.Compatible
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
f₁' : Z₁ ⟶ G.obj Z
f₂' : Z₂ ⟶ G.obj Z
H₁ : Presieve.functorPushforward G T f₁'
H₂ : Presieve.functorPushforward G T f₂'
eq : g₁ ≫ f₁' = g₂ ≫ f₂'
⊢ ℱ.val.map g₁.op (FamilyOfElements.functorPushforward G x f₁' H₁) =
    ℱ.val.map g₂.op (FamilyOfElements.functorPushforward G x f₂' H₂)

Proof state:
case mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : Sheaf K (Type w)
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : x.Compatible
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
X₁ : C
f₁ : X₁ ⟶ Z
h₁ : Z₁ ⟶ G.obj X₁
hf₁ : T f₁
H₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)
X₂ : C
f₂ : X₂ ⟶ Z
h₂ : Z₂ ⟶ G.obj X₂
hf₂ : T f₂
H₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)
eq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂
⊢ ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂)

Tactic:
apply hG.compatible ℱ h _ _ hf₁ hf₂

Result:
case mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : Sheaf K (Type w)
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : x.Compatible
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
X₁ : C
f₁ : X₁ ⟶ Z
h₁ : Z₁ ⟶ G.obj X₁
hf₁ : T f₁
H₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)
X₂ : C
f₂ : X₂ ⟶ Z
h₂ : Z₂ ⟶ G.obj X₂
hf₂ : T f₂
H₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)
eq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂
⊢ (g₁ ≫ h₁) ≫ G.map f₁ = (g₂ ≫ h₂) ≫ G.map f₂