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
X₂ : Arrow C₂
⊢ (adj₂.homEquiv (pushout.inl ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)) ≫
        ((F.leibnizPushout.obj X₁ ⋙ G.leibnizPullback.obj (op X₁)).obj X₂).hom) ≫
      pullback.fst ((G.obj (op (unop (op X₁)).left)).map ((F.leibnizPushout.obj X₁).obj X₂).hom)
        ((G.map (unop (op X₁)).hom.op).app ((F.leibnizPushout.obj X₁).obj X₂).right) =
    (((𝟭 (Arrow C₂)).obj X₂).hom ≫
        pullback.lift (adj₂.homEquiv (pushout.inr ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)))
          (adj₂.homEquiv (𝟙 ((F.obj (unop (op X₁)).right).obj ((𝟭 (Arrow C₂)).obj X₂).right))) ⋯) ≫
      pullback.fst ((G.obj (op (unop (op X₁)).left)).map ((F.leibnizPushout.obj X₁).obj X₂).hom)
        ((G.map (unop (op X₁)).hom.op).app ((F.leibnizPushout.obj X₁).obj X₂).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
X₂ : Arrow C₂
⊢ (adj₂.homEquiv (pushout.inl ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)) ≫
        ((F.leibnizPushout.obj X₁ ⋙ G.leibnizPullback.obj (op X₁)).obj X₂).hom) ≫
      pullback.snd ((G.obj (op (unop (op X₁)).left)).map ((F.leibnizPushout.obj X₁).obj X₂).hom)
        ((G.map (unop (op X₁)).hom.op).app ((F.leibnizPushout.obj X₁).obj X₂).right) =
    (((𝟭 (Arrow C₂)).obj X₂).hom ≫
        pullback.lift (adj₂.homEquiv (pushout.inr ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)))
          (adj₂.homEquiv (𝟙 ((F.obj (unop (op X₁)).right).obj ((𝟭 (Arrow C₂)).obj X₂).right))) ⋯) ≫
      pullback.snd ((G.obj (op (unop (op X₁)).left)).map ((F.leibnizPushout.obj X₁).obj X₂).hom)
        ((G.map (unop (op X₁)).hom.op).app ((F.leibnizPushout.obj X₁).obj X₂).right)

Tactic:
· simp [← homEquiv_naturality_one, ← homEquiv_naturality_two, pushout.condition]

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
X₂ : Arrow C₂
⊢ (adj₂.homEquiv (pushout.inl ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)) ≫
        ((F.leibnizPushout.obj X₁ ⋙ G.leibnizPullback.obj (op X₁)).obj X₂).hom) ≫
      pullback.snd ((G.obj (op (unop (op X₁)).left)).map ((F.leibnizPushout.obj X₁).obj X₂).hom)
        ((G.map (unop (op X₁)).hom.op).app ((F.leibnizPushout.obj X₁).obj X₂).right) =
    (((𝟭 (Arrow C₂)).obj X₂).hom ≫
        pullback.lift (adj₂.homEquiv (pushout.inr ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)))
          (adj₂.homEquiv (𝟙 ((F.obj (unop (op X₁)).right).obj ((𝟭 (Arrow C₂)).obj X₂).right))) ⋯) ≫
      pullback.snd ((G.obj (op (unop (op X₁)).left)).map ((F.leibnizPushout.obj X₁).obj X₂).hom)
        ((G.map (unop (op X₁)).hom.op).app ((F.leibnizPushout.obj X₁).obj X₂).right)