-- Module: Mathlib.CategoryTheory.LiftingProperties.ParametrizedAdjunction
Goal:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst : Category.{v₁, u₁} C₁
inst_1 : Category.{v₂, u₂} C₂
inst_2 : Category.{v₃, u₃} C₃
F : C₁ ⥤ Functor.{v₂, v₃, u₂, u₃} C₂ C₃
G : C₁ᵒᵖ ⥤ Functor.{v₃, v₂, u₃, u₂} C₃ C₂
adj₂ : F ⊣₂ G
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
X₃ Y₃ : C₃
f₃ : X₃ ⟶ Y₃
sq₁₂ : F.PushoutObjObj f₁ f₂
sq₁₃ : G.PullbackObjObj f₁ f₃
β : Arrow.mk f₂ ⟶ Arrow.mk sq₁₃.π
h₂ :
  Arrow.Hom.right
        ((fun (α : Arrow.mk sq₁₂.ι ⟶ Arrow.mk f₃) ↦
            (Arrow.homMk
                (((adj₂.homEquiv :
                      Equiv.{v₃ + 1, v₂ + 1} ((F.obj Y₁).obj (Arrow.mk f₂).left ⟶ X₃)
                        ((Arrow.mk f₂).left ⟶ (G.obj (op Y₁)).obj X₃)) :
                    (a : (F.obj Y₁).obj (Arrow.mk f₂).left ⟶ X₃) → (Arrow.mk f₂).left ⟶ (G.obj (op Y₁)).obj X₃)
                  (sq₁₂.inl ≫ Arrow.Hom.left α))
                (⋯.lift
                  (((adj₂.homEquiv :
                        Equiv.{v₃ + 1, v₂ + 1} ((F.obj X₁).obj (Arrow.mk f₂).right ⟶ X₃)
                          ((Arrow.mk f₂).right ⟶ (G.obj (op X₁)).obj X₃)) :
                      (a : (F.obj X₁).obj (Arrow.mk f₂).right ⟶ X₃) → (Arrow.mk f₂).right ⟶ (G.obj (op X₁)).obj X₃)
                    (sq₁₂.inr ≫ Arrow.Hom.left α))
                  (((adj₂.homEquiv :
                        Equiv.{v₃ + 1, v₂ + 1} ((F.obj Y₁).obj (Arrow.mk f₂).right ⟶ Y₃)
                          ((Arrow.mk f₂).right ⟶ (G.obj (op Y₁)).obj Y₃)) :
                      (a : (F.obj Y₁).obj (Arrow.mk f₂).right ⟶ Y₃) → (Arrow.mk f₂).right ⟶ (G.obj (op Y₁)).obj Y₃)
                    (Arrow.Hom.right α))
                  ⋯)
                ⋯ :
              Arrow.mk f₂ ⟶ Arrow.mk sq₁₃.π))
          ((fun (β : Arrow.mk f₂ ⟶ Arrow.mk sq₁₃.π) ↦
              (Arrow.homMk
                  (⋯.desc
                    (((Equiv.symm adj₂.homEquiv :
                          Equiv.{v₂ + 1, v₃ + 1} (X₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).left)
                            ((F.obj Y₁).obj X₂ ⟶ (Arrow.mk f₃).left)) :
                        (a : X₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).left) → (F.obj Y₁).obj X₂ ⟶ (Arrow.mk f₃).left)
                      (Arrow.Hom.left β))
                    (((Equiv.symm adj₂.homEquiv :
                          Equiv.{v₂ + 1, v₃ + 1} (Y₂ ⟶ (G.obj (op X₁)).obj (Arrow.mk f₃).left)
                            ((F.obj X₁).obj Y₂ ⟶ (Arrow.mk f₃).left)) :
                        (a : Y₂ ⟶ (G.obj (op X₁)).obj (Arrow.mk f₃).left) → (F.obj X₁).obj Y₂ ⟶ (Arrow.mk f₃).left)
                      (Arrow.Hom.right β ≫ sq₁₃.fst))
                    ⋯)
                  (((Equiv.symm adj₂.homEquiv :
                        Equiv.{v₂ + 1, v₃ + 1} (Y₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).right)
                          ((F.obj Y₁).obj Y₂ ⟶ (Arrow.mk f₃).right)) :
                      (a : Y₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).right) → (F.obj Y₁).obj Y₂ ⟶ (Arrow.mk f₃).right)
                    (Arrow.Hom.right β ≫ sq₁₃.snd))
                  ⋯ :
                Arrow.mk sq₁₂.ι ⟶ Arrow.mk f₃))
            β)) ≫
      sq₁₃.fst =
    Arrow.Hom.right β ≫ sq₁₃.fst
h₃ :
  Arrow.Hom.right
        ((fun (α : Arrow.mk sq₁₂.ι ⟶ Arrow.mk f₃) ↦
            (Arrow.homMk
                (((adj₂.homEquiv :
                      Equiv.{v₃ + 1, v₂ + 1} ((F.obj Y₁).obj (Arrow.mk f₂).left ⟶ X₃)
                        ((Arrow.mk f₂).left ⟶ (G.obj (op Y₁)).obj X₃)) :
                    (a : (F.obj Y₁).obj (Arrow.mk f₂).left ⟶ X₃) → (Arrow.mk f₂).left ⟶ (G.obj (op Y₁)).obj X₃)
                  (sq₁₂.inl ≫ Arrow.Hom.left α))
                (⋯.lift
                  (((adj₂.homEquiv :
                        Equiv.{v₃ + 1, v₂ + 1} ((F.obj X₁).obj (Arrow.mk f₂).right ⟶ X₃)
                          ((Arrow.mk f₂).right ⟶ (G.obj (op X₁)).obj X₃)) :
                      (a : (F.obj X₁).obj (Arrow.mk f₂).right ⟶ X₃) → (Arrow.mk f₂).right ⟶ (G.obj (op X₁)).obj X₃)
                    (sq₁₂.inr ≫ Arrow.Hom.left α))
                  (((adj₂.homEquiv :
                        Equiv.{v₃ + 1, v₂ + 1} ((F.obj Y₁).obj (Arrow.mk f₂).right ⟶ Y₃)
                          ((Arrow.mk f₂).right ⟶ (G.obj (op Y₁)).obj Y₃)) :
                      (a : (F.obj Y₁).obj (Arrow.mk f₂).right ⟶ Y₃) → (Arrow.mk f₂).right ⟶ (G.obj (op Y₁)).obj Y₃)
                    (Arrow.Hom.right α))
                  ⋯)
                ⋯ :
              Arrow.mk f₂ ⟶ Arrow.mk sq₁₃.π))
          ((fun (β : Arrow.mk f₂ ⟶ Arrow.mk sq₁₃.π) ↦
              (Arrow.homMk
                  (⋯.desc
                    (((Equiv.symm adj₂.homEquiv :
                          Equiv.{v₂ + 1, v₃ + 1} (X₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).left)
                            ((F.obj Y₁).obj X₂ ⟶ (Arrow.mk f₃).left)) :
                        (a : X₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).left) → (F.obj Y₁).obj X₂ ⟶ (Arrow.mk f₃).left)
                      (Arrow.Hom.left β))
                    (((Equiv.symm adj₂.homEquiv :
                          Equiv.{v₂ + 1, v₃ + 1} (Y₂ ⟶ (G.obj (op X₁)).obj (Arrow.mk f₃).left)
                            ((F.obj X₁).obj Y₂ ⟶ (Arrow.mk f₃).left)) :
                        (a : Y₂ ⟶ (G.obj (op X₁)).obj (Arrow.mk f₃).left) → (F.obj X₁).obj Y₂ ⟶ (Arrow.mk f₃).left)
                      (Arrow.Hom.right β ≫ sq₁₃.fst))
                    ⋯)
                  (((Equiv.symm adj₂.homEquiv :
                        Equiv.{v₂ + 1, v₃ + 1} (Y₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).right)
                          ((F.obj Y₁).obj Y₂ ⟶ (Arrow.mk f₃).right)) :
                      (a : Y₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).right) → (F.obj Y₁).obj Y₂ ⟶ (Arrow.mk f₃).right)
                    (Arrow.Hom.right β ≫ sq₁₃.snd))
                  ⋯ :
                Arrow.mk sq₁₂.ι ⟶ Arrow.mk f₃))
            β)) ≫
      sq₁₃.snd =
    Arrow.Hom.right β ≫ sq₁₃.snd
⊢ Arrow.Hom.right
      ((fun (α : Arrow.mk sq₁₂.ι ⟶ Arrow.mk f₃) ↦
          (Arrow.homMk
              (((adj₂.homEquiv :
                    Equiv.{v₃ + 1, v₂ + 1} ((F.obj Y₁).obj (Arrow.mk f₂).left ⟶ X₃)
                      ((Arrow.mk f₂).left ⟶ (G.obj (op Y₁)).obj X₃)) :
                  (a : (F.obj Y₁).obj (Arrow.mk f₂).left ⟶ X₃) → (Arrow.mk f₂).left ⟶ (G.obj (op Y₁)).obj X₃)
                (sq₁₂.inl ≫ Arrow.Hom.left α))
              (⋯.lift
                (((adj₂.homEquiv :
                      Equiv.{v₃ + 1, v₂ + 1} ((F.obj X₁).obj (Arrow.mk f₂).right ⟶ X₃)
                        ((Arrow.mk f₂).right ⟶ (G.obj (op X₁)).obj X₃)) :
                    (a : (F.obj X₁).obj (Arrow.mk f₂).right ⟶ X₃) → (Arrow.mk f₂).right ⟶ (G.obj (op X₁)).obj X₃)
                  (sq₁₂.inr ≫ Arrow.Hom.left α))
                (((adj₂.homEquiv :
                      Equiv.{v₃ + 1, v₂ + 1} ((F.obj Y₁).obj (Arrow.mk f₂).right ⟶ Y₃)
                        ((Arrow.mk f₂).right ⟶ (G.obj (op Y₁)).obj Y₃)) :
                    (a : (F.obj Y₁).obj (Arrow.mk f₂).right ⟶ Y₃) → (Arrow.mk f₂).right ⟶ (G.obj (op Y₁)).obj Y₃)
                  (Arrow.Hom.right α))
                ⋯)
              ⋯ :
            Arrow.mk f₂ ⟶ Arrow.mk sq₁₃.π))
        ((fun (β : Arrow.mk f₂ ⟶ Arrow.mk sq₁₃.π) ↦
            (Arrow.homMk
                (⋯.desc
                  (((Equiv.symm adj₂.homEquiv :
                        Equiv.{v₂ + 1, v₃ + 1} (X₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).left)
                          ((F.obj Y₁).obj X₂ ⟶ (Arrow.mk f₃).left)) :
                      (a : X₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).left) → (F.obj Y₁).obj X₂ ⟶ (Arrow.mk f₃).left)
                    (Arrow.Hom.left β))
                  (((Equiv.symm adj₂.homEquiv :
                        Equiv.{v₂ + 1, v₃ + 1} (Y₂ ⟶ (G.obj (op X₁)).obj (Arrow.mk f₃).left)
                          ((F.obj X₁).obj Y₂ ⟶ (Arrow.mk f₃).left)) :
                      (a : Y₂ ⟶ (G.obj (op X₁)).obj (Arrow.mk f₃).left) → (F.obj X₁).obj Y₂ ⟶ (Arrow.mk f₃).left)
                    (Arrow.Hom.right β ≫ sq₁₃.fst))
                  ⋯)
                (((Equiv.symm adj₂.homEquiv :
                      Equiv.{v₂ + 1, v₃ + 1} (Y₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).right)
                        ((F.obj Y₁).obj Y₂ ⟶ (Arrow.mk f₃).right)) :
                    (a : Y₂ ⟶ (G.obj (op Y₁)).obj (Arrow.mk f₃).right) → (F.obj Y₁).obj Y₂ ⟶ (Arrow.mk f₃).right)
                  (Arrow.Hom.right β ≫ sq₁₃.snd))
                ⋯ :
              Arrow.mk sq₁₂.ι ⟶ Arrow.mk f₃))
          β)) =
    Arrow.Hom.right β

Tactic:
apply sq₁₃.isPullback.hom_ext

Premises used: CategoryTheory.IsPullback.hom_ext, CategoryTheory.Functor.PullbackObjObj.isPullback