Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj

Goal state before:
pp: case h₀
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ : Arrow C₁
inst✝¹ : HasPullbacks C₂
inst✝ : HasPushouts C₃
X₃ : Arrow C₃
⊢ pushout.inl ((F.map X₁.hom).app ((G.leibnizPullback.obj (op X₁)).obj X₃).left)
        ((F.obj X₁.left).map ((G.leibnizPullback.obj (op X₁)).obj X₃).hom) ≫
      pushout.desc (adj₂.homEquiv.symm (𝟙 ((G.leibnizPullback.obj (op X₁)).obj X₃).left))
          (adj₂.homEquiv.symm
            (pullback.fst ((G.obj (op (unop (op X₁)).left)).map X₃.hom) ((G.map (unop (op X₁)).hom.op).app X₃.right)))
          ⋯ ≫
        ((𝟭 (Arrow C₃)).obj X₃).hom =
    pushout.inl ((F.map X₁.hom).app ((G.leibnizPullback.obj (op X₁)).obj X₃).left)
        ((F.obj X₁.left).map ((G.leibnizPullback.obj (op X₁)).obj X₃).hom) ≫
      ((G.leibnizPullback.obj (op X₁) ⋙ F.leibnizPushout.obj X₁).obj X₃).hom ≫
        adj₂.homEquiv.symm
          (pullback.snd ((G.obj (op (unop (op X₁)).left)).map X₃.hom) ((G.map (unop (op X₁)).hom.op).app X₃.right))

pp: case h₁
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ : Arrow C₁
inst✝¹ : HasPullbacks C₂
inst✝ : HasPushouts C₃
X₃ : Arrow C₃
⊢ pushout.inr ((F.map X₁.hom).app ((G.leibnizPullback.obj (op X₁)).obj X₃).left)
        ((F.obj X₁.left).map ((G.leibnizPullback.obj (op X₁)).obj X₃).hom) ≫
      pushout.desc (adj₂.homEquiv.symm (𝟙 ((G.leibnizPullback.obj (op X₁)).obj X₃).left))
          (adj₂.homEquiv.symm
            (pullback.fst ((G.obj (op (unop (op X₁)).left)).map X₃.hom) ((G.map (unop (op X₁)).hom.op).app X₃.right)))
          ⋯ ≫
        ((𝟭 (Arrow C₃)).obj X₃).hom =
    pushout.inr ((F.map X₁.hom).app ((G.leibnizPullback.obj (op X₁)).obj X₃).left)
        ((F.obj X₁.left).map ((G.leibnizPullback.obj (op X₁)).obj X₃).hom) ≫
      ((G.leibnizPullback.obj (op X₁) ⋙ F.leibnizPushout.obj X₁).obj X₃).hom ≫
        adj₂.homEquiv.symm
          (pullback.snd ((G.obj (op (unop (op X₁)).left)).map X₃.hom) ((G.map (unop (op X₁)).hom.op).app X₃.right))

Tactic:
· simp [← homEquiv_symm_naturality_two, ← homEquiv_symm_naturality_three]

Goal state after:
case h₁
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ : Arrow C₁
inst✝¹ : HasPullbacks C₂
inst✝ : HasPushouts C₃
X₃ : Arrow C₃
⊢ pushout.inr ((F.map X₁.hom).app ((G.leibnizPullback.obj (op X₁)).obj X₃).left)
        ((F.obj X₁.left).map ((G.leibnizPullback.obj (op X₁)).obj X₃).hom) ≫
      pushout.desc (adj₂.homEquiv.symm (𝟙 ((G.leibnizPullback.obj (op X₁)).obj X₃).left))
          (adj₂.homEquiv.symm
            (pullback.fst ((G.obj (op (unop (op X₁)).left)).map X₃.hom) ((G.map (unop (op X₁)).hom.op).app X₃.right)))
          ⋯ ≫
        ((𝟭 (Arrow C₃)).obj X₃).hom =
    pushout.inr ((F.map X₁.hom).app ((G.leibnizPullback.obj (op X₁)).obj X₃).left)
        ((F.obj X₁.left).map ((G.leibnizPullback.obj (op X₁)).obj X₃).hom) ≫
      ((G.leibnizPullback.obj (op X₁) ⋙ F.leibnizPushout.obj X₁).obj X₃).hom ≫
        adj₂.homEquiv.symm
          (pullback.snd ((G.obj (op (unop (op X₁)).left)).map X₃.hom) ((G.map (unop (op X₁)).hom.op).app X₃.right))